Idempotents in Group Algebras and Applications to Units

Vrije Universiteit Brussel - Faculty of Science and Bio-engineering Sciences

Group representations: idempotents in group algebras and applications to units

Graduation thesis submitted in fulﬁllment of the requirements for the degree of Doctor in Sciences

Author: Promotor: Inneke Van Gelder Prof. Dr. Eric Jespers Copromotor: Prof. Dr. Gabriela Olteanu

march 2015

ACKNOWLEDGMENTS

Ik zou graag iedereen willen bedanken die, elk op zijn manier, heeft bijgedragen tot de realisatie van deze doctoraatsthesis. Eerst en vooral zou ik graag mijn promotor Eric Jespers willen bedanken. Hij heeft zijn enthousiasme voor groepsringen op mij overgedragen door zijn interessante cursussen en boeiende lezingen. Door zijn ruime kennissenkring kwam ik in contact met verschillende vriendelijke wiskundigen waar ik veel van heb kunnen leren. I owe a very special thank you to my copromotor Gabriela Olteanu. I would like to thank her for her hospitality during my stays in Romania, for her support, both professional and personal. Thanks for the nice cooperations and the very careful proofreading of our joint works. Also my gratitude is expressed to my co-authors Andreas Bächle, Mauricio Caicedo, Angel´ del R´ıo,Florian Eisele and Ann Kiefer for the pleasant and enriching collaborations. Thanks to Allen Herman and Alexander Konovalov for the very interesting discussions about GAP and wedderga. Vervolgens zou ik ook graag mijn collega’s Andreas, Ann, Mauricio en Sara willen bedanken voor het nalezen van en hun kritische opmerkingen op een eerste versie van mijn doctoraatsthesis. Graag zou ik alle collega’s van de Vrije Universiteit Brussel, en in het bijzonder Philippe, Sara, Ann, Karen en Timmy, willen bedanken voor de fijne lunches, ontspannende koffiepauzes, toffe babbels, filmavonden, wiskundige nevenactiviteiten en zoveel meer! Verder ben ik ook dank verschuldigd aan het Fonds voor Wetenschappelijk Onderzoek om mij gedurende vier jaar financieel te ondersteunen. Hierbij moet ik ook Stefaan Caenepeel bedanken om mij gedurende een jaar een assistenten- positie aan te bieden in afwachting van een aanstelling door het FWO. Mijn familie en vrienden wil ik bedanken om mij onvoorwaardelijk te steunen. Mijn laatste, maar zeker niet de minste dank gaat uit naar mijn man Giel, om zijn steun en vertrouwen. Ook een heel dikke dankjewel om dit werk van een mooie omslag te voorzien.

Inneke Van Gelder maart 2015

CONTENTS contents i introduction iii summary ix publications xix samenvatting (summary in dutch) xxi list of notations xxxiii

1 preliminaries 1 1.1 Fixed point free groups ...... 2 1.2 Quaternion algebras ...... 3 1.3 Normal bases ...... 3 1.4 Number ﬁelds ...... 4 1.5 Crossed products ...... 10 1.6 Group rings ...... 11 1.7 Wedderburn-Artin decomposition ...... 14 1.8 Z-orders ...... 24 1.9 Congruence Subgroup Problem ...... 26 1.10 Finite subgroups of exceptional simple algebras ...... 31 1.11 Cyclotomic units ...... 33 1.12 Bass units ...... 34 1.13 Bicyclic units ...... 36 1.14 Central units ...... 39

2 wedderburn decomposition and idempotents 41 2.1 The Wedderburn decomposition of FG ...... 41 2.2 Primitive idempotents of QG ...... 55 2.3 Primitive idempotents of FG ...... 61 2.4 Conclusions ...... 66

i contents

3 exceptional components 67 3.1 Group algebras with exceptional components of type EC2 . . . 68 3.2 Group algebras with exceptional components of type EC1 . . . 72 3.3 Examples ...... 93

4 central units 101 4.1 Abelian groups ...... 101 4.1.1 A new proof of the Bass-Milnor Theorem ...... 103 4.1.2 A virtual basis of Bass units ...... 107 4.2 Strongly monomial groups ...... 110 4.3 Abelian-by-supersolvable groups ...... 113 4.3.1 Generalizing the Jespers-Parmenter-Sehgal Theorem . . 114 4.3.2 Reducing to a basis of products of Bass units ...... 120 4.4 Another class within the strongly monomial groups ...... 124 4.5 Conclusions ...... 133

5 applications to units of group rings 135 5.1 A subgroup of ﬁnite index in U(Z(Cqm o1 Cpn )) ...... 135 5.2 A method to compute U(ZG) up to commensurability . . . . . 137 5.3 Examples ...... 142 + 5.3.1 U(ZD16) up to ﬁnite index ...... 142 5.3.2 U(ZSL(2, 5)) up to commensurability ...... 143 5.4 Conclusions ...... 144 bibliography 147 index 155

ii INTRODUCTION

The notion of a group algebra already appeared in a paper of Arthur Cayley from 1854. However, only after the inﬂuential works of Richard Brauer (1901- 1977) and Emmy Noether (1882-1935) on representation theory, the subject gained attention because of the correspondence between modules of group algebras and group representations. In 1940, Graham Higman posed the following question in his Ph.D. thesis, for ﬁnite groups G and H:

Does ZG ' ZH imply that G ' H? This problem is referred to as the (integral) isomorphism problem. It was anticipated for a long time for this conjecture to be true. In 1987, Klaus W. Roggenkamp and Leonard L. Scott showed that this indeed is the case if G is a nilpotent group. It was a surprise when Martin Hertweck gave a counter example to the isomorphism problem in his Ph.D. thesis in 1998. Nowadays, it is still an important problem to decide for which classes of groups the conjecture does hold. In all these investigations, the unit group U(ZG) of ZG plays a fundamental role. It is essential to consider ZG as a Z-order in the (semisimple) rational group algebra QG and to have a detailed understanding of the Wedderburn decomposition of QG. If one proves the equality of two numbers a and b by showing ﬁrst that ‘a is less than or equal to b’ and then ‘a is greater than or equal to b’, it is unfair. One should instead show that they are really equal by disclosing the inner ground for their equality — Emmy Noether

The Wedderburn-Artin Theorem states that a semisimple ring R is isomorphic to a product of finitely many ni-by-ni matrix rings over division rings Di, for some integers ni. However, in the mindset of Emmy Noether, such a classification is unfair. One should instead aim to construct an explicit isomorphism between R and the product of matrix rings. To do this, a first important step is to calculate the primitive central idempotents e of R to distinguish the different matrix rings. Secondly, one needs to construct elements in each component Re, which play the role of a complete set of matrix units. In particular, one has to construct a complete set of orthogonal primitive idempotents.

iii introduction

A classical method for obtaining the primitive central idempotents of a semisimple group algebra FG involves computations using the irreducible characters of G over an algebraic closure of F . However, the known methods to compute the character table of a finite group are very time consuming. Therefore, in practical applications, the classical description of primitive central idempotents sometimes is of limited use. One would like a character-free description that only depends on the lattice of subgroups and the characteristic of the field, i.e. a description completely within FG. Such a description has been obtained by Aurora Olivieri, Angel´ del R´ıoand Juan Jacobo Simónin 2004 for the primitive central idempotents of QG when G is a strongly monomial group, for example an abelian-by-supersolvable group. This method relies on pairs of subgroups (H,K) of G satisfying some conditions which can be checked in- side the rational group algebra QG. Such pairs are called strong Shoda pairs of G. It turns out that each primitive central idempotent is the sum of the distinct conjugates of ε(H,K) (corresponding to a natural idempotent in the rational group algebra Q(H/K)) for a strong Shoda pair (H,K), which we denote by e(G, H, K). Furthermore, each simple component in the Wedderburn decomposition is a matrix ring over a crossed product of the finite abelian group NG(K)/H over a specific cyclotomic field for some strong Shoda pair (H,K). In 2007, Osnel Broche and Angel´ del R´ıotransfered those results to the case of semisimple finite group algebras FG for strongly monomial groups G. For arbitrary semisimple group algebras FG, it remains an open problem to give a character-free description of the primitive central idempotents and the Wedderburn decomposition of FG. For a rational group algebra QG of a finite nilpotent group G, a complete set of matrix units of an arbitrary simple component QGe(G, H, K) was given, in 2012, by Eric Jespers, Gabriela Olteanu and Angel´ del R´ıo.In joined work with Gabriela Olteanu, we gave a similar result for semisimple finite group algebras FG of nilpotent groups G. Moreover, examples were given to show that the method can not be extended to, for example, finite metacyclic groups. Chapter 1 is a preliminary chapter. In Chapter 2, we first study the primitive central idempotents and the Wedderburn decomposition of group algebras FG with F a number field and G a strongly monomial group (Theorem 2.1.6). This is a generalization of the results of Aurora Olivieri, Angel´ del R´ıo,Juan Jacobo Simónand Osnel Broche. Next, we focus on a complete set of matrix units in the Wedderburn components of QG and FG, with F a finite field, for a class of finite strongly monomial groups containing some metacyclic groups (Theorems 2.2.1 and 2.3.4).

iv introduction

I have never done anything ‘useful’. No discovery of mine has made, or is likely to make, directly or indirectly, for good or ill, the least diﬀerence to the amenity of the world — Godfrey Harold Hardy

Regardless of his sayings, much of the work of Godfrey Harold Hardy (1877- 1947) did find applications in different branches of science, other than mathematics. Hardy was a number theorist and exactly number theory is the elected area in pure mathematics to have many applications to other areas, such as coding theory and internet security. In 1974, Donald Knuth formulated this as follows: “Virtually every theorem in elementary number theory arises in a natural, motivated way in connection with the problem of making computers do high-speed numerical calculations”. Finite group algebras and their Wedderburn decomposition have applications to coding theory as well. Cyclic codes can be realized as ideals of group algebras of cyclic groups and many other important codes appear as ideals of group algebras of non-cyclic groups, see Section 2.3 for references. A concrete realization of the Wedderburn decomposition also allows applications to many other topics, for example to the investigation of the group of automorphisms of group rings, as shown by Au- rora Olivieri, Angel´ del R´ıoand Juan Jacobo Simónin 2006. In this thesis, we focus on the applications to the group of units of RG, where R is the ring of integers of a number field F . The main example is the group of units of integral group rings. Only for very few finite non-commutative groups G, a presentation of the group U(ZG) is known. However, Carl Ludwig Siegel, Armand Borel and Harish-Chandra showed, in a much more general setting, that U(RG) is always finitely generated, if G is finite. Therefore, we are satisfied with finding finitely many generators of U(RG), and in particular of U(ZG). If E is a complete collection of primitive central idempotents of FG, then M M RG ⊆ RGe ⊆ F Ge = FG e∈E e∈E

and each F Ge ' Mne (De) for some integers ne and some division rings De. L Since both RG and e∈E RGe are Z-orders in FG, we know that U(RG) is L of ﬁnite index in e∈E U(RGe). If we choose an order Oe in each De, then

also GLne (Oe) and U(RGe) have a common subgroup which is of finite index in both. This means that first, we have to find generating sets of units in

GLne (Oe), which is generated (up to ﬁnite index) by SLne (Oe) and the matrices with diagonal entries in U(Z(Oe)). So, the problem reduces to describing

SLne (Oe) and U(Z(Oe)).

v introduction

In Chapter 3, we classify the finite groups G such that, for a fixed abelian number field F , for all Wedderburn components Mn(D) in the group algebra FG, the corresponding SLn(O), for any Z-order O in D, is generated by the elementary matrices over a two-sided ideal in O (Theorems 3.1.2 and 3.2.21). The components Mn(D) where this is not possible are the so-called exceptional components. This investigation is a generalization of a result from Mauricio Caicedo and Angel´ del R´ıowho dealt with QG. It involves deep results from Hyman Bass, Bernhard Liehl, Leonid N. Vaserˇste˘ınand Tyakal Nanjundiah Venkataramana related to the Congruence Subgroup Problem. In Chapter 4, we study the central units Z(U(ZG)) for finite groups G. Due to Hyman Bass and John Willard Milnor (1966) it is well known that, for a finite abelian group G, the Bass units of the integral group ring ZG generate a subgroup of finite index in U(ZG). We give a new constructive proof of this result (Proposition 4.1.1). For non-abelian groups, some constructions of central units of ZG have been given by Eric Jespers, Guilherme Leal, Michael M. Parmenter, Sudarshan Sehgal and Raul Antonio Ferraz. This was done mainly for finite nilpotent groups G. We construct generalized Bass units and show that they generate a subgroup of finite index in Z(U(ZG)) for finite strongly monomial groups G (Theorem 4.2.3). For a class within the finite abelian-by-supersolvable groups G, we can do more and describe a multiplicatively independent set (based on Bass units) which generate a subgroup of finite index in Z(U(ZG)) (Theorem 4.3.8). For another class of finite strongly monomial groups containing some metacyclic groups, we construct such a set of multiplicatively independent elements starting from generalized Bass units (Theorem 4.4.4). In Chapter 5, we combine the results of the previous chapters to construct a generating set of U(ZG) up to finite index. This work is a continuation of a result from Eric Jespers, Gabriela Olteanu and Angel´ del R´ıofrom 2012, that described the unit group of ZG up to finite index for finite nilpotent groups G. We also continue works of JürgenRitter and Sudarshan Sehgal, and Eric Jespers and Guilherme Leal who showed that under some conditions the Bass units together with the bicyclic units generate a subgroup of finite index in U(ZG). If QG does not contain exceptional components, if one can construct matrix units in each Wedderburn component of QG and moreover, if one knows a generating set of Z(U(ZG)), then it is possible to describe U(ZG) up to finite index. We demonstrate this for metacyclic groups Cqm o1 Cpn , for different prime numbers p and q (Theorem 5.1.1). However, if QG has only exceptional components of type M2(D), then it turns out that SL2(O) can still

vi introduction

be generated by elementary matrices for a special (i.e. left norm Euclidean) Z- order O of D (Proposition 5.2.1). This allows us to construct the group of units of ZG up to ﬁnite index for ﬁnite groups G, such that QG has only exceptional components of one type and such that one knows non-central idempotents in the non-commutative non-exceptional components of QG (Proposition 5.2.2). Those non-central idempotents are needed to imitate the elementary matrices with (generalized) bicyclic units in ZG.

vii

SUMMARY

In this summary, we present our main results. For the convenience of the reader, Chapter 1 is devoted to a preliminary ex- position on quaternion algebras, number fields, crossed products, group rings, Z-orders, cyclotomic units, Bass units and bicyclic units. In Chapter 2, we give a concrete realization of the Wedderburn decomposition of group algebras FG of finite strongly monomial groups G over number fields F . This description is mainly based on the fact that, for rational group algebras QG of finite strongly monomial groups G, the Wedderburn decomposition is completely described using strong Shoda pairs.

Corollary 2.1.7 [8] If G is a ﬁnite strongly monomial group and F is a number ﬁeld, then every primitive central idempotent of FG is of the form eC (G, H, K) for a strong Shoda pair (H,K) of G and C ∈ CF (H/K). Furthermore, for every strong Shoda pair (H,K) of G and every C ∈ CF (H/K),

σ F GeC (G, H, K) ' M[G:E] F ζ[H:K] ∗τ E/H ,

where E = EF (G, H/K) and σ and τ are deﬁned as follows. Let yK be a generator of H/K and ψ : E/H → E/K be a left inverse of the projection E/K → E/H. Then

i ψ(gH) i σgH (ζk) = ζk, if yK = y K, 0 j 0 −1 0 j τ(gH, g H) = ζk, if ψ(gg H) ψ(gH)ψ(g H) = y K,

for gH, g0H ∈ E/H and integers i and j.

Next, we obtain more information on the Wedderburn decomposition of QG and determine a complete set of orthogonal primitive idempotents in each component determined by a strong Shoda pair provided the twisting τ is trivial.

ix summary

Theorem 2.2.1 [3] Let (H,K) be a strong Shoda pair of a finite group G such that the 0 0 twisting τ(gH, g H) = 1 for all g, g ∈ NG(K). Let ε = ε(H,K) and e = e(G, H, K). Let F denote the fixed subfield of QHε under the natural action of NG(K)/H and [NG(K): H] = n. Let w be a normal element of QHε/F and B the normal basis determined by w. Let ψ be the F -isomorphism between QNG(K)ε and the matrix algebra Mn(F ) with respect to the basis B determined as follows:

ψ : QNG(K)ε = QHε ∗ NG(K)/H → Mn(F ) 0 xuσ 7→ [x ◦ σ]B,

0 for x ∈ QHε, σ ∈ Gal(QHε/F ) ' NG(K)/H, where x denotes multiplication by x on QHε. Let P,A ∈ Mn(F ) be deﬁned as follows:  1 1 1 ··· 1 1   0 0 ··· 0 1   1 −1 0 ··· 0 0   1 0 ··· 0 0       1 0 −1 ··· 0 0   0 1 ··· 0 0  P =  ......  and A =  . . . . .  .  ......   . . . . .   ......   . . . . .   1 0 0 · · · −1 0   0 0 ··· 0 0  1 0 0 ··· 0 −1 0 0 ··· 1 0

Then −1 {xTc1εx : x ∈ T2 hxei} is a complete set of orthogonal primitive idempotents of QGe where we set −1 −1 xe = ψ (P AP ), T1 is a transversal of H in NG(K) and T2 is a right 1 P transversal of NG(K) in G. By T1 we denote the element t in c |T1| t∈T1 QG. We apply this result in Corollary 2.2.5 to all metacyclic groups of the form Cqm o1 Cpn , with p and q different prime numbers. We finish the chapter with a translation of the above theorem to finite semisimple group algebras (Theorem 2.3.4, [6]). In Chapter 3, we classify finite groups G and abelian number fields F such that FG contains an exceptional component in its Wedderburn decomposition. Hyman Bass (1964), Leonid N. Vaserˇste˘ın(1973), Bernhard Liehl (1981), Tyakal Nanjundiah Venkataramana (1994) and Ernst Kleinert (2000) showed that, under some conditions, the elementary matrices En(I) for all non-zero ideals I in any order O in a finite dimensional rational division algebra gen-

x summary

erate a subgroup of ﬁnite index in SLn(O). More precisely, if a matrix ring Mn(D) over a ﬁnite dimensional rational division ring D is not of one of the following forms:

n = 1 and D is a non-commutative division ring other than a totally deﬁnite quaternion algebra;

n = 2 and D equals Q, a quadratic imaginary extension of Q, or a totally deﬁnite quaternion algebra with center Q,

then [SLn(O): En(I)] < ∞ for any order O in D and any non-zero ideal I in O. A simple finite dimensional rational algebra is called exceptional if it is in the list above. The exceptional simple algebras occurring as Wedderburn components of a group algebra, are very restricted. Corollary 1.9.9 [7] If a simple finite dimensional rational algebra is an exceptional component of some group algebra FG for some number field F , then it is of one of the following types:

EC1: a non-commutative division ring other than a totally deﬁnite quaternion algebra;

√ √ √ Ä −1,−1 ä EC2: M2( ), M2( ( −1)), M2( ( −2)), M2( ( −3)), M2 , Q Q Q Q Q Ä −1,−3 ä Ä −2,−5 ä M2 , M2 . Q Q

We first classify all exceptional components of type EC2 occurring in the Wedderburn decomposition of group algebras of finite groups over arbitrary number fields. We do this by giving a full list of finite groups G, number fields F and exceptional components M2(D) such that M2(D) is a faithful Wedderburn component of FG. Theorem 3.1.2 [9] Let F be a number field, G be a finite group and B a simple exceptional algebra of type EC2. Then B is a faithful Wedderburn component of FG if and only if G, F , B is a row listed in Table 2 on page 70.

Secondly, we classify F -critical groups, i.e. groups G such that FG has an exceptional component of type EC1 in its Wedderburn decomposition, but

xi summary

no proper quotient has this property. Note that any group H such that FH has a non-commutative division ring (not totally deﬁnite quaternion) in its Wedderburn decomposition has an epimorphic F -critical image G such that if an exceptional component D of type EC1 appears as a faithful Wedderburn component of FG, then also FH has D as a simple component.

Theorem 3.2.21 [9] Let D be a division ring and F an abelian number ﬁeld, p and q diﬀerent odd prime numbers. Then D is a Wedderburn component of FG for an F -critical group G if and only if one of the following holds:

−1,−1 (a) D = F , G ∈ {SL(2, 3),Q8}, F is totally imaginary and both, e2(F/Q) and f2(F/Q), are odd;

Ä −1,−1 ä (b) D = , G ∈ {SL(2, 3)×Cp,Q8 ×Cp}, gcd(p, |G|/p) = 1, op(2) F (ζp) is odd, F is totally real and both, e2(F (ζp)/Q) and f2(F (ζp)/Q), are odd;

−1 2 −1,(ζp−ζp ) (c) D = −1 , G = Cp o2 C4, p ≡ −1 mod 4, F totally F (ζp+ζp ) −1 imaginary, Q(ζp)∩F ⊆ Q(ζp +ζp ) and both, ep(F/Q) and fp(F/Q), are odd;

−1 2 −1,(ζp−ζp ) (d) D = −1 , G = Cq × (Cp o2 C4), p ≡ −1 mod 4, oq(p) F (ζq ,ζp+ζp ) odd, F is totally real and both, ep(F (ζq)/Q) and fp(F (ζq)/Q), are odd;

n (e) D = (K(ζp)/K, σ, ζk) with Schur index k , G = haip ok hbin with −1 r n n ≥ 8, gcd(p, n) = 1, b ab = a , and both k and k are divisible n −1 r r k by all the primes dividing n. Here K = F (ζk, ζp + ζp + ... + ζp ) r and σ : F (ζpk) → F (ζpk): ζp 7→ ζp ; ζk 7→ ζk. Moreover Q(ζp) ∩ F ⊆ n −1 r r k Q(ζp + ζp + ... + ζp ) and one of the conditions (i) - (iii) from Theorem 3.2.20 holds. Furthermore

® f ´ p − 1 k n min l ∈ N ≡ 0 mod = gcd(pf − 1, e) gcd(k, l) k

with e = ep(F (ζpk)/K) and f = fp(K/Q).

xii summary

Essential here is to use the classification of finite subgroups of division rings by Shimshon Avraham Amitsur and the classification of maximal finite subgroups of 2 × 2-matrices over totally definite quaternion algebras with center Q given by Gabriele Nebe. In Chapter 4, we investigate the group of central units of ZG. First, we give a new constructive proof for the famous Bass-Milnor result avoiding K-theory. Additionally, we construct a virtual basis in the unit group of ZG for finite abelian groups G.

Corollary 4.1.6 [4] Let G be a ﬁnite abelian group. For every cyclic subgroup C of G, choose a generator aC of C and for every k coprime to the order of C, choose an m integer mk,C with k k,C ≡ 1 mod |C|. Then

ß |C| ™ u (a ): C cyclic subgroup of G, 1 < k < , gcd(k, |C|) = 1 k,mk,C C 2

is a virtual basis of U(ZG). Moreover, for any Bass unit uk,m(g) in ZG we have u (g)c = h u (a )n0 u (a )n1 , k,m k0,mk0,C C k1,mk1,C C

for C = hgi, an element h ∈ G and integers c, n0, n1, k0, k1 such that |C| ±k1 1 ≤ k0, k1 ≤ 2 , g = aC and k0 ≡ ±kk1 mod |C|.

For ﬁnite non-abelian groups G, we restrict to strongly monomial groups because of the detailed description of QG in this case.

Theorem 4.2.3 [5] Let G be a ﬁnite strongly monomial group. The group generated by the n 0 0 0 generalized Bass units b G,H , with b = uk,m(1 − Hc + hHc) for a strong Shoda pair (H,K) of G, h ∈ H and nG,H0 the minimal positive integer n 0 such that b G,H ∈ ZG, contains a subgroup of ﬁnite index in Z(U(ZG)).

Since we know the rank of Z(U(ZG)), we know a priori the number of elements in a virtual basis of Z(U(ZG)).

xiii summary

Theorem 4.2.1 [3] Let G be a ﬁnite strongly monomial group. The rank of Z(U(ZG)) equals Ç å X φ([H : K]) − 1 , k(H,K)[NG(K): H] (H,K)

where (H,K) runs through a complete and non-redundant set of strong Shoda pairs of G, h is such that H = hh, Ki and

ß 1 if hhn ∈ K for some n ∈ N (K); k = G (H,K) 2 otherwise.

Let u ∈ U(Z hgi), for g ∈ G. Consider a subnormal series

N : N0 = hgi ¡ N1 ¡ N2 ¡ ··· ¡ Nm = G. N We deﬁne c0 (u) = u and N Y N h ci (u) = ci−1(u) ,

h∈Ti where Ti is a transversal for Ni in Ni−1, and prove that this construction behaves well. Deﬁne

l Sg = {l ∈ U(Z/|g|Z): g is conjugate with g in G}

and denote Sg = hSg, −1i. This construction yields a virtual basis of the group Z(U(ZG)) in the following setting. Theorem 4.3.8 [5] Let G be a ﬁnite abelian-by-supersolvable group, such that every cyclic subgroup of order not a divisor of 4 or 6, is subnormal in G. Let R denote a set of representatives of Q-classes of G. For g ∈ R, choose a transversal Tg of Sg in U(Z/|g|Z) containing 1 and for every k ∈ Tg \{1} choose an m integer mk,g with k k,g ≡ 1 mod |g|. For every g ∈ R of order not a divisor of 4 or 6, choose a subnormal series Ng from hgi to G, which is normalized by NG(hgi). Then

Ng c (uk,mk,g (g)) : g ∈ R, k ∈ Tg \{1}

is a virtual basis of Z(U(ZG)).

xiv summary

Next, we focus on another subclass of the ﬁnite strongly monomial groups. Let H be a ﬁnite group and K a subgroup of H such that H/K = hgKi is a cyclic group of order pn. Let k be a positive integer coprime with p and let r be an arbitrary integer. For every 0 ≤ s ≤ n, we set

s cs(H, K, k, r) = 1 and, for 0 ≤ j ≤ s − 1, we construct recursively the following products of generalized Bass units of ZH:

Ö èps−j−1

s Y rpn−s cj (H, K, k, r) = uk,opn (k)nH,K (g hK“ + 1 − K“) h∈hgpn−j ,Ki

Ñ s−1 é j−1 ! Y s −1 Y s+l−j −1 cl (H, K, k, r) cl (H, K, k, r) . l=j+1 l=0

Theorem 4.4.4 [3] Let G be a ﬁnite strongly monomial group such that there exists a complete and non-redundant set S of strong Shoda pairs (H,K) of G, with the property that each [H : K] is a prime power. For every (H,K) ∈ S, let TK be a right transversal of NG(K) in G, let I(H,K) be a set of representatives of U(Z/[H : K]Z) modulo hNG(K)/H, −1i containing 1 and let [H : K] = n(H,K) p(H,K) , with p(H,K) a prime number. The set    Y Y n(H,K) t  c0 (H, K, k, x) :(H,K) ∈ S, k ∈ I(H,K) \{1}

t∈TK x∈NG(K)/H 

is a virtual basis of Z(U(ZG)).

The class of groups mentioned in Theorem 4.4.4, contains the metacyclic groups Cqm o1 Cpn and we apply our result in Corollary 4.4.5. In Chapter 5, we ﬁrst apply Corollaries 2.2.5 and 4.4.5 to construct explicitly generators for three nilpotent subgroups of U(ZG) that together generate a subgroup of ﬁnite index in U(Z(Cqm o1 Cpn )).

xv summary

Theorem 5.1.1 [3]

Let p and q be different prime numbers. Let G = Cqm o1 Cpn be a finite metacyclic group with Cpn = hbi and Cqm = hai. Assume that either ¨ qj ∂ q 6= 3, or n 6= 1 or p 6= 2. For every j = 1, . . . , m, let Kj = a , let Fj be the center of QGε(hai ,Kj), fix a normal element wj of Q(ζqj )/Fj and let ψj be the Fj-isomorphism between QGε(hai ,Kj) and the matrix algebra Mpn (Fj) with respect to the normal basis Bj associated to wj, determined as follows:

ψj : QGε(hai ,Kj) = Q hai ε(hai ,Kj) ∗ G/ hai → Mpn (Fj) 0 xuσ 7→ [x ◦ σ]Bj ,

0 for x ∈ Q hai ε(hai ,Kj), σ ∈ G/ hai, where x denotes multiplication by x −1 −1 −1 on Q hai ε(hai ,Kj). Let xj = ψj (P )bε(hai ,Kj)ψj (P ) , with

 1 1 1 ··· 1 1   1 −1 0 ··· 0 0     1 0 −1 ··· 0 0  P =  ......  ,  ......   ......   1 0 0 · · · −1 0  1 0 0 ··· 0 −1

k n and tj be a positive integer such that tjxj ∈ ZG for all k with 1 ≤ k ≤ p . The following two groups are ﬁnitely generated nilpotent subgroups of U(ZG): ≠ ∑ + n 2 h −k ﬂhbi n Vj = 1 + p tj yxj bbxj : y ∈ hai , h, k ∈ {1, . . . , p }, h < k ,

≠ ∑ − n 2 h −k ﬂhbi n Vj = 1 + p tj yxj bbxj : y ∈ hai , h, k ∈ {1, . . . , p }, h > k .

+ Qm + − Qm − Hence V = j=1 Vj and V = j=1 Vj are nilpotent subgroups of U(ZG). Furthermore, the group

U, V +,V − ,

with U as in Corollary 4.4.5, is of ﬁnite index in U(ZG).

xvi summary

Next, we generalize results of JürgenRitter and Sudarshan Sehgal, and of Eric Jespers and Guilherme Leal who developed many classes of finite groups in which U(ZG) is generated up to finite index by the Bass units (denoted B1(G)) and the bicyclic units (denoted B2(G)). The exceptions are the finite groups G such that their rational group algebra QG has exceptional components or such that G has non-abelian fixed point free homomorphic images. Proposition 5.2.2 [7] Ln Ln Let G be a finite group and let QG = i=1 QGei ' i=1 Mni (Di) be the Wedderburn decomposition of QG. Assume that QG does not contain exceptional components of type EC1. Also, assume that for each integer i ∈ {1, . . . , n} such that ni 6= 1 and QGei is not exceptional (of type EC2), Gei is not fixed point free. For every exceptional component QGei ' M2(Di), Di has a left norm Euclidean order Oi. Take a Z-basis Bi of Oi and let ψi : M2(Di) → QGei be a Q-algebra isomorphism. For such i, set ß Å 0 x ã Å 0 0 ã ™ U := 1 + ψ , 1 + ψ : x ∈ B . i i 0 0 i x 0 i S The subgroup U := hB1(G) ∪ B2(G) ∪ i Uii of QG is commensurable with U(ZG).

+ To finish this thesis, we demonstrate our technique on the group D16 and the fixed point free group SL(2, 5). As far as we are aware, this is the first technique known to describe the unit group of ZSL(2, 5) up to commensurability.

xvii

PUBLICATIONS

All results presented in this document have appeared previously (partially, identically or modiﬁed) in the following publications:

[1] Gabriela Olteanu and Inneke Van Gelder. Finite group algebras of nilpotent groups: a complete set of orthogonal primitive idempotents. Finite Fields Appl., 17(2):157–165, 2011.

[2] Osnel Broche, Allen Herman, Alexander Konovalov, Aurora Olivieri, Gabriela Olteanu, Angel´ del R´ıo, and Inneke Van Gelder. Wedderga - Wedderburn Decomposition of Group Algebras. Version 4.5.1+, 2013. www.cs.st-andrews.ac.uk/ alexk/wedderga.

[3] Eric Jespers, Gabriela Olteanu, Angel´ del R´ıo,and Inneke Van Gelder. Group rings of ﬁnite strongly monomial groups: central units and primitive idempotents. J. Algebra, 387:99–116, 2013.

[4] Eric Jespers, Angel´ del R´ıo, and Inneke Van Gelder. Writing units of integral group rings of ﬁnite abelian groups as a product of Bass units. Math. Comp., 83(285):461–473, 2014.

[5] Eric Jespers, Gabriela Olteanu, Angel´ del R´ıo,and Inneke Van Gelder. Central units of integral group rings. Proc. Amer. Math. Soc., 142(7):2193–2209, 2014.

[6] Gabriela Olteanu and Inneke Van Gelder. Construction of minimal non- abelian left group codes. Des. Codes Cryptogr., 2014. doi:10.1007/s10623- 014-9922-z.

[7] Florian Eisele, Ann Kiefer, and Inneke Van Gelder. Describing units of integral group rings up to commensurability. J. Pure Appl. Algebra, 219(7):2901–2916, 2015.

[8] Gabriela Olteanu and Inneke Van Gelder. On idempotents and the number of simple components of semisimple group algebras. arXiv, abs/1411.5929. preprint.

xix publications

[9] Andreas Bächle, Mauricio Caicedo, and Inneke Van Gelder. A classification of exceptional components in group algebras over abelian number fields. arXiv, abs/1412.5458. preprint.

xx SAMENVATTING(SUMMARYINDUTCH)

In deze samenvatting schetsen we kort de geschiedenis van de studie van groepsringen en stellen we onze hoofdresultaten voor. In 1940 stelde Graham Higman de volgende vraag in zijn doctoraatsthesis, voor twee eindige groepen G en H:

Volgt uit het isomorfisme ZG ' ZH dat G ' H? Dit probleem is gekend als het (gehele) isomorfismeprobleem. Lange tijd werd ervan uitgegaan dat deze conjectuur waar was. In 1987 toonden Klaus W. Roggenkamp en Leonard L. Scott dat deze stelling inderdaad opgaat als G een eindige nilpotente groep is. Het kwam dan ook als een verrassing toen Martin Hertweck in 1998 in zijn doctoraatsthesis een tegenvoorbeeld gaf voor het isomorfismeprobleem. Desalniettemin is het vandaag de dag nog steeds een belangrijk probleem om te beslissen voor welke klassen van groepen de conjectuur wel geldig blijft. In dit onderzoek speelt de eenhedengroep U(ZG) een fundamentele rol. Hierin is het essentieel om ZG te beschouwen als een Z- order in de (semisimpele) rationale groepsalgebra QG en om een gedetailleerd inzicht te hebben in de Wedderburndecompositie van QG.

Als men de gelijkheid van twee getallen a en b bewijst door eerst te tonen dat ‘a kleiner is of gelijk aan b’ en dan ‘a groter is of gelijk aan b’, is dat oneerlijk. Men zou in plaats daarvan moeten tonen dat ze echt gelijk zijn door de diepere reden van hun gelijkheid te belichten — Emmy Noether

De Wedderburn-Artin Stelling stelt dat een semisimpele ring R isomorf is met een product van eindig veel ni × ni matrixringen over (scheve) lichamen Di, voor zekere natuurlijke getallen ni. In de gedachtegang van Emmy Noether is zo’n classiﬁcatie echter oneerlijk. In plaats daarvan moet men streven naar de constructie van een expliciet isomorﬁsme tussen R en het product van matrixringen. Een eerste belangrijke stap hiervoor, is het bepalen van primitieve centrale idempotenten e van R, die de verschillende matrixringen van elkaar onderscheiden. Ten tweede moet men elementen in elke component Re construeren die de rol spelen van een volledige verzameling matrixeenheden. In

xxi samenvatting (summary in dutch)

het bijzonder tracht men een volledige verzameling van orthogonale primitieve idempotenten te bepalen. Een klassieke methode voor het bepalen van primitieve centrale idempotenten in een semisimpele groepsalgebra FG brengt berekeningen met zich mee die gebruik maken van de irreduciebele karakters van G, over een algebra¨ısche sluiting van F . De gekende methoden om karaktertabellen van eindige groepen te berekenen zijn echter tijdrovend. Daarom is de klassieke beschrijving van primitieve centrale idempotenten soms slechts beperkt bruikbaar voor prak- tische toepassingen. Men verkiest een karaktervrije beschrijving die enkel af- hangt van de verzameling van deelgroepen en de karakteristiek van het lichaam, d.w.z. een beschrijving volledig in FG. Zo’n beschrijving werd bekomen door Aurora Olivieri, Angel´ del R´ıoen Juan Jacobo Sim´onin 2004 voor de primitieve centrale eenheden van QG, in het geval dat G een sterk monomiale groep is, bijvoorbeeld abels-bij-superoplosbaar. Deze methode steunt op paren van deelgroepen (H,K) van G die aan enkele voorwaarden voldoen. Deze paren noemen we sterke Shoda paren van G. Het blijkt dat elke primitieve centrale idempotent de som is van de verschillende geconjugeerden van ε(H,K) (een natuurlijk idempotent in de rationale groepsalgebra Q(H/K)), voor een sterk Shoda paar (H,K). We noteren dit element met e(G, H, K). Elke Wed- derburncomponent is bovendien een matrixring over een kruisproduct van de eindige abelse groep NG(K)/H, over een bepaald cyclotomisch lichaam voor een sterk Shoda paar (H,K). In 2007 vertaalden Osnel Broche en Angel´ del R´ıo deze resultaten naar het geval van semisimpele eindige groepsalgebra’s FG voor sterk monomiale groepen G. Voor willekeurige semisimple groepsalgebra’s FG blijft het een open probleem om een karaktervrije beschrijving van de primitieve centrale idempotenten en de Wedderburndecompositie van FG te geven. Voor de rationale groepsalgebra QG van een eindige nilpotente groep G beschreven Eric Jespers, Gabriela Olteanu en Angel´ del R´ıoin 2012 een volledige verzameling van matrixeenheden in een willekeurige enkelvoudige component QGe(G, H, K). In samenwerking met Gabriela Olteanu gaven we een gelijk- aardig resultaat voor semisimpele eindige groepsalgebra’s FG voor nilpotente groepen G. Bovendien tonen voorbeelden aan dat de methode niet kan uitge- breid worden naar, bijvoorbeeld, eindige metacyclische groepen. In Hoofdstuk 1 geven we een inleiding tot quaternionenalgebra’s, getallenlichamen, kruisproducten, groepsringen, Z-orders, cyclotomische eenheden, Bass eenheden en bicyclische eenheden.

xxii samenvatting (summary in dutch)

In Hoofdstuk 2 geven we een concrete realisatie van de Wedderburndecompo- sitie van groepsalgebra’s FG van eindige sterk monomiale groepen over getallenlichamen F . Deze beschrijving is hoofdzakelijk gebaseerd op de beschrijving van Aurora Olivieri, Angel´ del R´ıo,Juan Jacobo Sim´onen Osnel Broche en het feit dat voor rationale groepsalgebra’s QG van eindig sterk monomiale groepen G, de Wedderburndecompositie volledig bepaald is door sterke Shoda paren. Gevolg 2.1.7 [8] Zij G een eindige sterk monomiale groep en F een getallenlichaam. Dan is elke primitieve centrale idempotent van FG van de vorm eC (G, H, K) voor een sterk Shoda paar (H,K) van G en C ∈ CF (H/K). Bovendien, voor elk sterk Shoda paar (H,K) van G en elke C ∈ CF (H/K),

σ F GeC (G, H, K) ' M[G:E] F ζ[H:K] ∗τ E/H ,

waar E = EF (G, H/K) en σ en τ gedeﬁnieerd zijn als volgt. Zij yK een voortbrenger voor H/K en ψ : E/H → E/K een links inverse van de projectie E/K → E/H. Dan

i ψ(gH) i σgH (ζk) = ζk, als yK = y K, 0 j 0 −1 0 j τ(gH, g H) = ζk, als ψ(gg H) ψ(gH)ψ(g H) = y K,

voor gH, g0H ∈ E/H en natuurlijke getallen i en j.

Vervolgens verkrijgen we meer informatie over de Wedderburndecompositie van QG en bepalen we een volledige verzameling van orthogonale primitieve idempotenten in elke Wedderburncomponent die bepaald wordt door een sterk Shoda paar met een triviale afbeelding τ. Stelling 2.2.1 [3] Zij (H,K) een sterk Shoda paar van een eindige groep G zodat voor 0 0 alle g, g ∈ NG(K) geldt dat τ(gH, g H) = 1. Zij ε = ε(H,K) en e = e(G, H, K). Zij F het deellichaam van QHε dat invariant is onder de actie van NG(K)/H en stel [NG(K): H] = n. Zij w een normaal element van QHε/F en B de normale basis bepaald door w. Zij ψ het F -isomorﬁsme tussen QNG(K)ε en de matrixalgebra Mn(F ) ten opzichte van de basis B bepaald als volgt:

0 ψ : QNG(K)ε = QHε ∗ NG(K)/H → Mn(F ): xuσ 7→ [x ◦ σ]B,

xxiii samenvatting (summary in dutch)

0 voor x ∈ QHε, σ ∈ Gal(QHε/F ) ' NG(K)/H, waar x staat voor de vermenigvuldiging met x op QHε. Zij P,A ∈ Mn(F ) gedeﬁnieerd als volgt:

 1 1 1 ··· 1 1   0 0 ··· 0 1   1 −1 0 ··· 0 0   1 0 ··· 0 0       1 0 −1 ··· 0 0   0 1 ··· 0 0  P =  ......  en A =  . . . . .  .  ......   . . . . .   ......   . . . . .   1 0 0 · · · −1 0   0 0 ··· 0 0  1 0 0 ··· 0 −1 0 0 ··· 1 0 Dan is −1 {xTc1εx : x ∈ T2 hxei} een volledige verzameling van orthogonale primitieve idempotenten van −1 −1 QGe met xe = ψ (P AP ), T1 een transversaal van H in NG(K) en T2 een rechts transversaal van NG(K) in G. Met Tc1 noteren we het element 1 P t in G. |T1| t∈T1 Q

We passen onze resultaten toe in Gevolg 2.2.5 op alle metacyclische groepen van de vorm Cqm o1 Cpn , waarbij p en q verschillende priemgetallen zijn. We eindigen het hoofdstuk met een vertaling van bovenstaande resultaten naar semisimpele eindige groepsalgebra’s (Stelling 2.3.4, [6]).

Ik heb nooit iets ‘nuttigs’ gedaan. Geen enkele ontdekking van mij heeft direct of indirect ook maar de minste bijdrage geleverd, ten goede of ten kwade, aan de leefbaarheid van de wereld en zal dat waarschijnlijk ook nooit doen. — Godfrey Harold Hardy

Ongeacht zijn uitspraken vond het werk van Godfrey Harold Hardy (1877- 1947) wel toepassingen in verschillende takken van de wetenschap buiten de wiskunde. Hardy was een getaltheoreticus en getaltheorie is nu net het uit- gelezen gebied binnen de zuivere wiskunde om verscheidene toepassingen te hebben, denk maar aan codetheorie en internetbeveiliging. In 1974 formu- leerde Donald Knuth dit als volgt: “Zo goed als elke stelling in de elementaire getaltheorie staat op een natuurlijke manier in verband met het probleem om computers numerieke berekeningen aan hoge snelheid te laten uitvoeren”. Ook eindige groepsalgebra’s en hun Wedderburndecompositie hebben toepassingen binnen de codetheorie. Cyclische codes kunnen gerealiseerd worden als idealen in groepsalgebra’s van cyclische groepen en ook vele andere belangrijke codes

xxiv samenvatting (summary in dutch) verschijnen als idealen in groepsalgebra’s van niet-cyclische groepen, zie Sectie 2.3 voor referenties. Een concrete realisatie van de Wedderburndecompositie laat ook vele andere toepassingen toe, bijvoorbeeld het onderzoeken van de automorﬁsmengroep van groepsringen, zoals aangetoond door Aurora Olivieri, Angel´ del R´ıoen Juan Jacobo Sim´onin 2006. In deze thesis focussen we op de toepassingen voor de eenhedengroep van RG, met R de ring van gehele getallen van een getallenlichaam F . Het belangrijkste voorbeeld is de eenhedengroep van gehele groepsringen. Slechts voor zeer wei- nig eindige niet-abelse groepen G is een presentatie van de groep U(ZG) gekend. Nochtans bewezen Carl Ludwig Siegel, Armand Borel en Harish-Chandra, in een veel algemenere context, wel dat U(RG) altijd eindig voortgebracht is voor G een eindige groep. Daarom zijn we al tevreden met het vinden van eindig veel voortbrengers voor U(RG), en in het bijzonder voor U(ZG). Als E een volledige verzameling van primitieve centrale idempotenten van FG is, dan

M M RG ⊆ RGe ⊆ F Ge = FG e∈E e∈E

en elke F Ge ' Mne (De) voor bepaalde natuurlijke getallen ne en (scheve) L lichamen De. Vermits zowel RG als e∈E RGe een Z-order in FG is, weten L we dat U(RG) van eindige index is in e∈E U(RGe). Als we een Z-order Oe

in elke De kiezen, dan hebben GLne (Oe) en U(RGe) een gemeenschappelijke deelgroep die van eindige index in beide is. Dit betekent dat we in de eerste

plaats een voortbrengende verzameling moeten vinden voor GLne (Oe), die voortgebracht wordt (op eindige index na) door SLne (Oe) en de matrices met diagonale elementen in U(Z(Oe)). Het probleem wordt dus herleid tot het

beschrijven van SLne (Oe) en U(Z(Oe)). In Hoofdstuk 3, classiﬁceren we de eindige groepen G waarvoor, gegeven een willekeurig maar vast abels getallenlichaam F , voor alle Wedderburncomponen- ten Mn(D) in de groepsalgebra FG, de bijhorende SLn(O), voor elk Z-order O in D, voortgebracht is door de elementaire matrices over een tweezijdig ideaal in O. Dit onderzoek is een uitbreiding van een resultaat van Mauricio Caicedo en Angel´ del R´ıo(2014) en gaat terug tot diepe resultaten van Hy- man Bass (1964), Leonid N. Vaserˇste˘ın(1973), Bernhard Liehl (1981), Tyakal Nanjundiah Venkataramana (1994) en Ernst Kleinert (2000) gerelateerd aan het congruentiedeelgroepenprobleem. Beter gezegd, als een matrixring Mn(D) over een eindig dimensionaal rationaal (scheef) lichaam D niet van volgende vorm is:

xxv samenvatting (summary in dutch)

n = 1 en D is een niet-commutatief scheef lichaam verschillend van een totaal deﬁniete quaternionenalgebra;

n = 2 en D is gelijk aan Q, een kwadratische imaginaire uitbreiding van Q of een totaal deﬁniete quaternionenalgebra met centrum Q,

dan [SLn(O): En(I)] < ∞ voor elk order O in D en elk niet-nul ideaal I van O. De componenten Mn(D) van FG die wel voorkomen in de vorige lijst noemen we de exceptionele componenten. De exceptionele componenten die kunnen optreden als een Wedderburncomponent van een groepsalgebra zijn zeer beperkt. Gevolg 1.9.9 [7] Als een enkelvoudige eindig dimensionale rationale algebra een exceptionele component is van een groepsalgebra FG voor een getallenlichaam F , dan is de algebra van een van de volgende types:

EC1: een niet-commutatief scheef lichaam verschillend van een totaal de- ﬁniete quaternionenalgebra;

√ √ √ Ä −1,−1 ä EC2: M2( ), M2( ( −1)), M2( ( −2)), M2( ( −3)), M2 , Q Q Q Q Q Ä −1,−3 ä Ä −2,−5 ä M2 , M2 . Q Q

We classiﬁceren eerst alle exceptionele componenten van type EC2 in de Wedderburndecompositie van groepsalgebra’s van eindige groepen over willekeurige getallenlichamen. Dit doen we door een volledige lijst te geven van eindige groepen G, getallenlichamen F en exceptionele componenten M2(D) zodat M2(D) een getrouwe Wedderburncomponent is van FG. Stelling 3.1.2 [9] Zij F een getallenlichaam, G een eindige groep en B een enkelvoudige exceptionele algebra van type EC2. Dan is B een getrouwe Wedderburn- component van FG als en slechts als G, F en B een rij vormen in Tabel 2 op pagina 70.

Vervolgens classiﬁceren we F -kritische groepen, d.w.z. groepen G zodat FG een exceptionele component van type EC1 in zijn Wedderburndecompo- sitie bevat, maar geen enkel echt quoti¨ent deze eigenschap heeft. Merk op dat elke groep H waarvoor FH een niet-commutatief scheef lichaam (geen

xxvi samenvatting (summary in dutch) totaal deﬁniete quaternionenalgebra) in zijn Wedderburndecompositie bevat, een epimorf F -kritisch beeld G heeft.

Stelling 3.2.21 [9] Zij D een scheef lichaam, F een abels getallenlichaam en p en q verschillende oneven priemgetallen. Dan is D een Wedderburncomponent van FG voor een F -kritische groep G als en slechts als een van de volgende gevallen geldt:

−1,−1 (a) D = F , G ∈ {SL(2, 3),Q8}, F is totaal imaginair, e2(F/Q) en f2(F/Q) zijn oneven;

Ä −1,−1 ä (b) D = , G ∈ {SL(2, 3) × Cp,Q8 × Cp}, ggd(p, |G|/p) = 1, F (ζp) op(2) is oneven, F is totaal re¨eel en e2(F (ζp)/Q) en f2(F (ζp)/Q) zijn oneven;

−1 2 −1,(ζp−ζp ) (c) D = −1 , G = Cp o2 C4, p ≡ −1 mod 4, F is totaal F (ζp+ζp ) −1 imaginair, Q(ζp) ∩ F ⊆ Q(ζp + ζp ) en ep(F/Q) en fp(F/Q) zijn oneven;

−1 2 −1,(ζp−ζp ) (d) D = −1 , G = Cq × (Cp o2 C4), p ≡ −1 mod 4, oq(p) F (ζq ,ζp+ζp ) oneven, F is totaal re¨eel en ep(F (ζq)/Q) en fp(F (ζq)/Q) zijn oneven; n (e) D = (K(ζp)/K, σ, ζk) met Schur index k , G = haip ok hbin waar −1 r n n ≥ 8, ggd(p, n) = 1, b ab = a , zowel k als k zijn deelbaar door n −1 r r k alle priemdelers van n. Hier K = F (ζk, ζp + ζp + ... + ζp ) en r σ : F (ζpk) → F (ζpk): ζp 7→ ζp ; ζk 7→ ζk. Bovendien Q(ζp) ∩ F ⊆ n −1 r r k Q(ζp + ζp + ... + ζp ) en een van de voorwaarden (i) - (iii) uit Theorem 3.2.20 gelden. Ook

® f ´ p − 1 k n min l ∈ N ≡ 0 mod = ggd(pf − 1, e) ggd(k, l) k

met e = ep(F (ζpk)/K) en f = fp(K/Q).

Hier is het essentieel om gebruik te maken van de classiﬁcatie van eindige deelgroepen van scheve lichamen door Shimshon Avraham Amitsur en de classi-

xxvii samenvatting (summary in dutch)

ﬁcatie van maximale eindige deelgroepen in 2×2-matrices over totaal deﬁniete quaternionenalgebra’s met centrum Q door Gabriele Nebe. In Hoofdstuk 4, bestuderen we de centrale eenheden Z(U(ZG)) voor eindige groepen G. Eerst geven we een nieuw en constructief bewijs voor de bekende stelling van Hyman Bass en John Willard Milnor waarin we het gebruik van K-theorie vermijden. Bovendien construeren we een virtuele basis in de eenhedengroep van ZG voor eindige abelse groepen G. Gevolg 4.1.6 [4] Zij G een eindige abelse groep. Kies voor elke cyclische deelgroep C van G een voortbrenger aC van C en kies voor elke k relatief priem met de m orde van C een natuurlijk getal mk,C zodat k k,C ≡ 1 mod |C|. Dan is

ß |C| ™ u (a ): C cyclische deelgroep, 1 < k < , ggd(k, |C|) = 1 k,mk,C C 2

een virtuele basis van U(ZG). Bovendien geldt voor elke Bass eenheid uk,m(g) in ZG dat

u (g)c = h u (a )n0 u (a )n1 , k,m k0,mk0,C C k1,mk1,C C

±k1 voor C = hgi, een h ∈ G en gehele getallen c, n0, n1, k0, k1 zodat g = aC , |C| 1 ≤ k0, k1 ≤ 2 en k0 ≡ ±kk1 mod |C|.

Voor sommige niet-abelse groepen zijn constructies van centrale eenheden van ZG gegeven door Eric Jespers, Guilherme Leal, Michael M. Parmenter, Su- darshan Sehgal en Raul Antonio Ferraz. Dit werd gedaan vooral voor eindige nilpotente groepen G. Wij construeren veralgemeende Bass eenheden en tonen dat deze een deelgroep voortbrengen die van eindige index is in Z(U(ZG)), voor eindige sterk monomiale groepen G.

Stelling 4.2.3 [5] Zij G een eindige sterk monomiale groep. De groep voortgebracht door de n 0 0 0 veralgemeende Bass eenheden b G,H , met b = uk,m(1 − Hc + hHc) voor een sterk Shoda paar (H,K) van G, h ∈ H en nG,H0 minimaal zodat n 0 b G,H ∈ ZG, bevat een deelgroep van eindige index in Z(U(ZG)).

Aangezien we de rang van Z(U(ZG)) kennen, weten we op voorhand al exact hoeveel elementen er in een virtuele basis van Z(U(ZG)) moeten zitten. xxviii samenvatting (summary in dutch)

Stelling 4.2.1 [3] Zij G een eindige sterk monomiale groep. Dan is de rang van Z(U(ZG)) gelijk aan Ç å X φ([H : K]) − 1 , k(H,K)[NG(K): H] (H,K) waar (H,K) loopt doorheen een volledige en niet-redundante verzameling van sterke Shoda paren van G, h is zo dat H = hh, Ki en

ß 1 als hhn ∈ K voor een n ∈ N (K); k = G (H,K) 2 anders.

Zij u ∈ U(Z hgi), met g ∈ G. Beschouw een subnormale rij

N : N0 = hgi ¡ N1 ¡ N2 ¡ ··· ¡ Nm = G.

N We deﬁni¨eren c0 (u) = u en

N Y N h ci (u) = ci−1(u) ,

h∈Ti

met Ti een transversaal voor Ni in Ni−1, en we bewijzen dat deze constructie zich goed gedraagt. Deﬁnieer

l Sg = {l ∈ U(Z/|g|Z): g is geconjugeerd met g in G}

en noteer Sg = hSg, −1i. Deze constructie leidt tot een virtuele basis van Z(U(ZG)) op de volgende manier. Stelling 4.3.8 [5] Zij G een eindige abels-bij-superoplosbare groep zodat elke cyclische deelgroep, van orde niet gelijk aan een deler van 4 of 6, subnormaal is in G. Zij R een verzameling van representanten van Q-klassen van G. Kies een transversaal Tg van Sg in U(Z/|g|Z) die 1 bevat voor g ∈ R en kies voor m elke k ∈ Tg \{1} een natuurlijk getal mk,g, zodat k k,g ≡ 1 mod |g|. Kies voor elke g ∈ R, van orde niet gelijk aan een deler van 4 of 6, een subnormale rij Ng van hgi naar G, die genormaliseerd wordt door NG(hgi). Dan is Ng c (uk,mk,g (g)) : g ∈ R, k ∈ Tg \{1} een virtuele basis van Z(U(ZG)).

xxix samenvatting (summary in dutch)

Vervolgens concentreren we ons op een andere deelklasse van de eindige sterk monomiale groepen. Zij H een eindige groep en K een deelgroep van H zodat H/K = hgKi een cyclische groep is van orde pn. Zij k een natuurlijk getal relatief priem met p en zij r een willekeurig geheel getal. Voor elke 0 ≤ s ≤ n, s deﬁni¨eren we cs(H, K, k, r) = 1 en, voor 0 ≤ j ≤ s−1, construeren we recursief het volgende product van veralgemeende Bass eenheden van ZH: Ö èps−j−1

s Y rpn−s cj (H, K, k, r) = uk,opn (k)nH,K (g hK“ + 1 − K“) h∈hgpn−j ,Ki

Ñ s−1 é j−1 ! Y s −1 Y s+l−j −1 cl (H, K, k, r) cl (H, K, k, r) . l=j+1 l=0

Stelling 4.4.4 [3] Zij G een eindige sterk monomiale groep zodat er een volledige en niet- redundante verzameling S van sterke Shoda paren (H,K) van G bestaat zodat elke [H : K] een macht van een priemgetal is. Voor elke (H,K) ∈ S, zij TK een rechts transversaal van NG(K) in G, zij I(H,K) een verzameling representanten van U(Z/[H : K]Z) modulo hNG(K)/H, −1i die 1 bevat n(H,K) en zij [H : K] = p(H,K) , met p(H,K) een priemgetal. Dan is    Y Y n(H,K) t  c0 (H, K, k, x) :(H,K) ∈ S, k ∈ I(H,K) \{1}

t∈TK x∈NG(K)/H 

een virtuele basis van Z(U(ZG)).

De klasse groepen uit Stelling 4.4.4, bevat de eindige metacyclische groepen Cqm o1 Cpn en we passen het resultaat toe in Gevolg 4.4.5. In Hoofdstuk 5 combineren we de resultaten van de voorgaande hoofdstuk- ken om een voortbrengende verzameling van U(ZG) op eindige index na te construeren. Dit is een voortzetting van het werk van Eric Jespers, Gabriela Olteanu en Angel´ del R´ıouit 2012 waarin ze U(ZG) op eindige index na beschreven voor eindige nilpotente groepen. Als QG geen exceptionele componenten bevat, als men matrixeenheden in elke Wedderburncomponent van QG kan construeren en als men bovendien een voortbrengende verzameling van

xxx samenvatting (summary in dutch)

Z(U(ZG)) kent, dan is het mogelijk om U(ZG) te beschrijven op eindige index na. We demonstreren dit voor metacyclische groepen Cqm o1 Cpn met verschillende priemgetallen p en q. Stelling 5.1.1 [3]

Zij p en q verschillende priemgetallen. Zij G = Cqm o1 Cpn een eindige metacyclische groep waarbij Cpn = hbi en Cqm = hai. Onderstel dat ofwel ¨ qj ∂ q 6= 3, ofwel n 6= 1 of p 6= 2. Voor elke j = 1, . . . , m, stel Kj = a , stel Fj het centrum van QGε(hai ,Kj), kies een normaal element wj van Q(ζqj )/Fj en zij ψj het Fj-isomorﬁsme tussen QGε(hai ,Kj) en de matrixalgebra Mpn (Fj) ten opzichte van de normale basis Bj geassocieerd aan wj, bepaald als volgt:

ψj : QGε(hai ,Kj) = Q hai ε(hai ,Kj) ∗ G/ hai → Mpn (Fj) 0 xuσ 7→ [x ◦ σ]Bj , 0 met x ∈ Q hai ε(hai ,Kj), σ ∈ G/ hai, waar x staat voor de vermenigvul- −1 −1 −1 diging met x op Q hai ε(hai ,Kj). Zij xj = ψj (P )bε(hai ,Kj)ψj (P ) , met  1 1 1 ··· 1 1   1 −1 0 ··· 0 0     1 0 −1 ··· 0 0  P =  ......  ,  ......   ......   1 0 0 · · · −1 0  1 0 0 ··· 0 −1 k en tj natuurlijke getallen zodat tjxj ∈ ZG voor alle k met 1 ≤ k ≤ pn. Dan zijn de volgende twee groepen eindig voortgebrachte nilpotente deelgroepen van U(ZG): ≠ ∑ + n 2 h −k ﬂhbi n Vj = 1 + p tj yxj bbxj : y ∈ hai , h, k ∈ {1, . . . , p }, h < k , ≠ ∑ − n 2 h −k ﬂhbi n Vj = 1 + p tj yxj bbxj : y ∈ hai , h, k ∈ {1, . . . , p }, h > k .

+ Qm + − Qm − Bijgevolg zijn V = j=1 Vj en V = j=1 Vj nilpotente deelgroepen van U(ZG). Bovendien is de groep U, V +,V − ,

met U zoals in Gevolg 4.4.5 van eindige index in U(ZG).

xxxi samenvatting (summary in dutch)

Wanneer QG toch exceptionele componenten bevat, maar enkel van type EC2, dan blijkt dat SL2(O) nog steeds voortgebracht kan worden door elementaire matrices voor een speciaal (links norm Euclidisch) Z-order O van D. Dit zorgt ervoor dat we de eenhedengroep van ZG op eindige index na kunnen construeren voor eindige groepen G, zodat QG enkel exceptionele componenten heeft van type EC2 en zodanig dat men niet-centrale idempotenten kent in de niet-commutatieve niet-exceptionele componenten van QG. Deze niet- centrale idempotenten zijn nodig om de elementaire matrices te imiteren met (veralgemeende) bicyclische eenheden in ZG. Dit werk bouwt verder op resultaten van Jurgen¨ Ritter en Sudarshan Sehgal, en van Eric Jespers en Guilherme Leal die vele klassen van eindige groepen beschreven waar U(ZG) voorgebracht wordt door de Bass eenheden (genoteerd als B1(G)) en de bicyclische eenheden (genoteerd als B2(G)) op eindige index na. De uitzonderingen zijn de eindige groepen G zodat hun rationale groepsalgebra QG exceptionele componenten bevat of zodat G niet-abelse fixpuntvrije epimorfe beelden heeft. Stelling 5.2.2 [7] Ln Ln Zij G een eindige groep en zij QG = i=1 QGei ' i=1 Mni (Di) de Wedderburndecompositie van QG. Onderstel dat QG geen exceptionele componenten van type EC1 bevat. Stel ook dat voor elke i ∈ {1, . . . , n} waarvoor ni 6= 1 en QGei niet exceptioneel is (van type EC2), Gei niet fixpuntvrij is. Het scheef lichaam Di heeft een links norm Euclidisch order Oi voor elke exceptionele component QGei ' M2(Di). Neem een Z-basis Bi van Oi en zij ψi : M2(Di) → QGei een Q-algebra-isomorfisme. Voor zo’n i, definieer

ß Å 0 x ã Å 0 0 ã ™ U := 1 + ψ , 1 + ψ : x ∈ B . i i 0 0 i x 0 i S De deelgroep U := hB1(G) ∪ B2(G) ∪ i Uii van QG en U(ZG) bevatten een gezamenlijke deelgroep van eindige index in beiden.

+ Om de thesis te besluiten, demonstreren we onze techniek op de groep D16 en de ﬁxpuntvrije groep SL(2, 5). Voor zover we weten is dit de eerste gekende techniek om de eenhedengroep van ZSL(2, 5) te beschrijven.

xxxii LISTOFNOTATIONS

[G, H] commutator of G and H, page 1 hgin cyclic group generated by g of order n, page 1 |g| order of a group element g, page 1 −1 P H“ group ring element |H| h∈H h, page 2 gb h”gi, page 2

Ffp completion of F with respect to p, page 5 x›Y orbit sum, page 136 (L/F, τ) classical crossed product, page 10

Ä a,b ä F quaternion algebra over F , page 3

B1(G) group generated by the Bass units of ZG, page 35

B2(G) group generated by the bicyclic units of ZG, page 36

Cn cyclic group order n, page 1

CenG(α) centralizer of α in G, page 2

CF (G) set of orbits of the faithful characters of G under the action of Gal(F (ζ|G|)/F ), page 41

EF (G, H/K) stabilizer of any C ∈ CF (H/K) under the action of NG(H) ∩ NG(K), page 44

En(R) group generated by elementary matrices, page 27 e(G, H, K) P ε(H,K)t, page 16 t∈G/CenG(ε(H,K)) e (G, H, K) P ε (H,K)t, page 42 C t∈G/CenG(εC (H,K)) C ep(L/F ) ramiﬁcation index, page 7

xxxiii list of notations

F (χ) ﬁeld of character values, page 16

F G ﬁxed subﬁeld of F under G, page 2 fp(L/F ) residue degree, page 7

G0 commutator subgroup of G, page 1

G om H semidirect product of H acting on G with kernel of order m, page 1

GB2(G) group generated by the generalized bicyclic units of ZG, page 37

GLn(R) general linear group, page 26 gcd(r, m) greatest common divisor of r and m, page 1

H < GH is a proper subgroup of G, page 1

H ≤ GH is a subgroup of G, page 1

H ¡ GH is a proper normal subgroup of G, page 1

H ¢ GH is a normal subgroup of G, page 1 ind(A) Schur index, page 4

Mn(R) n × n-matrices, page 4 mp(A) p-local index, page 7

NG(H) normalizer of H in G, page 1 nrdA/K (a) reduced norm, page 26 om(r) multiplicative order of r modulo m, page 1

RG group ring, page 12

α R ∗τ G crossed product, page 10 S ⊂ TS is a proper subset of T , page 1

S ⊆ TS is a subset of T , page 1

xxxiv list of notations

SLn(R) special linear group, page 26

SLn(R, m) congruence subgroup of level m, page 26

uk,mnG,M (1 − Mc + gMc) generalized Bass unit, page 111 uk,m(g) Bass unit, page 34 U(R) unit group of R, page 25

k vp(m) maximum non-negative integer k such that p divides m, page 1

Z(G) center of G, page 1

βei,h generalized bicyclic unit, page 36

βg,h bicyclic unit, page 36 χG induced character, page 17

ε(H,H) H“, page 16 Q ε(H,K) M/K∈M(H/K)(K“ − Mc), page 16

−1 P P −1 εC (H,K) |H| h∈H ψ∈C ψ(hK)h , page 42

j ηk(ζn) cyclotomic unit, page 34

γei,h generalized bicyclic unit, page 37

γg,h bicyclic unit, page 36 φ Euler’s totient function, page 8

ζm complex primitive m-th root of unity, page 1

xxxv

PRELIMINARIES

One denotes the set of positive integers (without zero) by N. All integers are denoted by Z and the rational numbers by Q. Further, one refers to the real numbers by R and the complex numbers by C. For the integers modulo k, one uses the notation Z/kZ. For integers r and m, one uses gcd(r, m) to refer to the greatest common divisor of r and m. For integers r, m and p, with p a prime number and gcd(r, m) = 1, one deﬁnes

k vp(m) = maximum non-negative integer k such that p divides m;

om(r) = multiplicative order of r modulo m;

ζm = complex primitive m-th root of unity.

Throughout this thesis, G will be a finite group and F a field. One denotes by char(F ) the characteristic of F and by FG the group algebra of G over F . For two sets S and T , one writes S ⊆ T (resp. S ⊂ T ) if S is a subset (resp. proper subset) of T . The notation H ≤ G (resp. H < G, H ¢ G, H ¡ G) signifies that H is a subgroup (resp. proper subgroup, normal subgroup, proper normal subgroup) of G. For a group element g, one uses the notation |g| for its order. By Z(G), one denotes the center of G. When g is an element of a group

G with order n, one writes hgi or hgin to denote the cyclic group (of order n) generated by g. If no explicit generator is given, one writes Cn to refer to the cyclic group of order n. For subgroups G and H of a common group, one writes [G, H] to denote the commutator of G and H, i.e. the subgroup generated by all elements g−1h−1gh with g ∈ G and h ∈ H. For [G, G], i.e. the commutator 0 subgroup of G, one also uses the notation G . By G om H one denotes a semidirect product of H acting on G with kernel of order m. For H ≤ G, g ∈ G and h ∈ H, one writes Hg = g−1Hg and hg = g−1hg. Analogously, for g −1 α ∈ FG and g ∈ G, α = g αg. For H ≤ G, NG(H) denotes the normalizer

1 preliminaries

of H in G. When the characteristic of F does not divide the order of G, one −1 P sets H“ = |H| h∈H h, an idempotent of FG; if H = hgi, then one simply writes gb for h”gi. For each α ∈ FG, CenG(α) denotes the centralizer of α in G. Furthermore, if F is a field and G a group of automorphisms of F , one writes F G for the fixed subfield of F under G. In this work, the algebra FG, the ring RG, for the ring of integers R of a number field F , and in particular its unit group U(RG) play a prominent role. A fundamental problem is to find generators and determine the structure of U(RG), and in particular of U(ZG). This problem has already been widely studied by Bass, Sehgal, Ritter, Zassenhaus, Higman, Passman, Marciniak, Hoechsmann, Jespers, del R´ıo,Kimmerle, Parmenter and many others. The study of these objects combines methods from representation theory, K-theory, number theory and group theory. A complete review on the current state of the research of group rings and their unit groups is written by Eric Jespers and Angel´ del R´ıo[JdR]. In this chapter, for the convenience of the reader, we give a survey on the tools used in the investigation of U(RG).

1.1 fixed point free groups

A finite group G is said to be fixed point free if it has an (irreducible) complex representation ρ such that 1 is not an eigenvalue of ρ(g), for all 1 6= g ∈ G.A finite group G is said to be a Frobenius group if it contains a proper non-trivial subgroup H such that H ∩ Hg = {1}, for all g ∈ G \ H. The group H is called a Frobenius complement in G. Fixed point free groups are exactly the Frobenius complements, as explained in the book of Donald S. Passman [Pas68, Theorem 18.1.v].

Theorem 1.1.1 A finite group G is fixed point free if and only if G is a Frobenius complement (in some other finite group).

Hans Zassenhaus described all Frobenius complements and in particular proved the following result. A proof can be found in [Pas68, Theorem 18.6].

Theorem 1.1.2 (Zassenhaus) SL(2, 5) is the smallest non-solvable Frobenius complement.

2 1.2 quaternion algebras

1.2 quaternion algebras

Ä a,b ä Given a ﬁeld F , as well as two elements a, b ∈ F , the quaternion algebra F is: Åa, bã F hi, ji = , F (i2 = a, j2 = b, ij = −ji) where F hi, ji denotes the free F -algebra on the non-commuting free variables i and j. The following proposition determines when a quaternion algebra is a division ring. A proof can be found in a book by Richard S. Pierce [Pie82, Proposition 1.6].

Proposition 1.2.1 Ä a,b ä Let F be a ﬁeld with char(F ) 6= 2. The quaternion algebra F is a division ring if and only if the equation ax2 + by2 = z2 does not have a non-zero solution (x, y, z) in F 3.

Ä a,b ä A quaternion algebra F is said to be totally deﬁnite if F is contained in C, is totally real and a and b are totally negative.

1.3 normal bases

Let K be a ﬁnite Galois extension of a ﬁeld F . Often, one is interested in an F -basis of K. The Normal Basis Theorem answers this need. It can be found in many standard references, see for example [Art44, Theorem 28].

Theorem 1.3.1 (Normal Basis Theorem)

Let K be a ﬁnite normal extension of a ﬁeld F and let σ1, . . . , σn be the elements in the Galois group of K over F . There always exists an element w ∈ K such that the elements σ1(w), . . . , σn(w) are linearly independent with respect to F .

If K/F is a ﬁnite Galois extension, then there exists an element w ∈ K such that {σ(w): σ ∈ Gal(K/F )} is an F -basis of K, a so-called normal basis, whence w is called normal in K/F .

3 preliminaries

1.4 number fields

We recall some basic background on number ﬁelds and local Schur indices, mainly based on the books of Edwin Weiss [Wei98] and Irving Reiner [Rei75].

For a positive integer n and a ring R, one denotes by Mn(R) the n × n- matrices over R.A central simple algebra over a field F is a finite dimensional associative algebra A, which is simple, and for which the center is exactly F . The dimension of a central simple algebra A as a vector space over its center is always a square; the degree of A is the square root of this dimension. A field F is called a number field if F is a finite extension of Q. Let F be a number field and A a finite dimensional central simple F -algebra. A famous result of Joseph Wedderburn states that A = Mn(D) for some division algebra D with center F (see for example [Rei75, Theorem 7.4]). The Schur index of A, denoted by ind(A), is the degree of D. Clearly, A is itself a division algebra if and only if ind(A) equals the degree of A. A valuation of a field F is a function ϕ from F into the non-negative reals such that for all a, b ∈ F :

1. ϕ(a) = 0 if and only if a = 0;

2. ϕ(ab) = ϕ(a)ϕ(b);

3. ϕ(a + b) ≤ ϕ(a) + ϕ(b).

If the valuation also satisﬁes the stronger condition ϕ(a+b) ≤ max{ϕ(a), ϕ(b)}, then ϕ is said to be non-archimedean. Otherwise, one says that ϕ is archimedean.

A valuation ϕ of a ﬁeld F is always associated to a topology Tϕ on F . For each a ∈ F , a fundamental system of neighborhoods of a is given by the set of all U(a, ε) = {b ∈ F : ϕ(a − b) < ε},

for ε > 0, Two valuations ϕ and ψ are equivalent if for all a ∈ F , ϕ(a) ≤ 1 if and only if ψ(a) ≤ 1. Equivalent valuations determine the same topology. From now on we assume F to be a number ﬁeld. One way of obtaining archimedean valuations is the following. The ordinary absolute value | · | is an archimedean valuation on C. Let σ be an embedding of F in C, then |σ(a)|, a ∈ F , deﬁnes an archimedean valuation on F . In particular, if F has r

4 1.4 number fields

embeddings in R and s pairs of complex embeddings in C, then one can obtain in this way r + s archimedean valuations. Now we consider some non-archimedean valuations. Let R be the ring of integers of F and p a non-zero prime ideal of R. For each non-zero a ∈ F , the principal ideal aR factors into a product of powers of prime ideals. Let ep(a) denote the exponent to which p occurs in this factorization. If p does not occur, we set ep(a) = 0. Also, ep(0) = +∞. Fix 0 < ε < 1 and set vp(0) = 0 and ep(a) vp(a) = ε for a ∈ F \{0}. In this case, vp is a non-archimedean valuation of F , called the p-adic valuation on F . The “Big” Ostrowski Theorem states that every archimedean valuation of F is equivalent to exactly one of the r + s valuations arising from embeddings into C and that any non-archimedean valuation of F is equivalent to a p-adic valuation for a unique prime ideal p in the ring of integers of F . The “Little” Ostrowski Theorem deals with the case F = Q and is well known (see for example [Pie82, Theorem 17.3]). The general case is folklore and explained nicely by Keith Conrad in one of his many handouts [Con]. A prime of F is an equivalence class of valuations of F . One excludes the trivial valuation ϕ defined by ϕ(0) = 0, ϕ(a) = 1 for a ∈ F \{0}. Since F is a number field, there are the infinite primes of F arising from embeddings of F into C and the finite primes of F , arising from p-adic valuations of F , with p ranging over the distinct prime ideals in the ring of integers of F . In many references the primes of F are also called places.

Let ϕ be a valuation of F . A sequence (an)n∈N, an ∈ F , is called a ϕ- Cauchy sequence if am −an converge to 0 in the topology Tϕ when both m and n converges to +∞. One says that F is complete if every ϕ-Cauchy sequence converges. By definition, F‹ (or (F,‹ ϕe)) is the completion of F with respect to ϕ if F is dense in F‹, the valuation ϕ extends to a valuation ϕe on F‹ and (F,‹ ϕe) is complete. If ϕ is an archimedean valuation, then the completion of F is either R or C and in each case ϕe is equivalent to the ordinary absolute value. The completion of F , with respect to the p-adic valuation vp on F , will be denoted by Ffp. One defines the local Schur index (or local index) of a central simple F -algebra A at p as mp(A) = ind(Ffp ⊗F A). Example 1.4.1 If F = Q, then its ring of integers is Z and the non-zero prime ideals of Z are exactly the ideals pZ with p ranging over the prime numbers. Hence the finite primes of Q correspond to the prime numbers in N. The completion of Q at a prime p is the set Qp of p-adic numbers. Furthermore,

5 preliminaries

an inﬁnite prime of Q corresponds to the unique embedding of Q in R and is often simply denoted by ∞.

The following theorem (see [Rei75, Theorems 31.8 and 32.11]) states that division rings with a fixed center are uniquely determined up to isomorphism by reduced fractions modulo Z. Theorem 1.4.2 Let F be a number field and denote by Pl(F ) the set of all primes (finite and infinite) of F . There exists an injective map from the isomorphism classes of division rings with center F to the sequences of reduced fractions modulo Z in (Q/Z)(Pl(F )). Moreover, this map is defined such that, if Å r ã D 7→ p , mp p∈Pl(F )

then the denominators equal the local Schur indices mp = mp(D) at the primes p ∈ Pl(F ).

For a ﬁxed prime p of F , the reduced fraction rp , associated to a division mp ring D by the injective map from Theorem 1.4.2, is known as the local Hasse invariant at p of D. The following theorem is a well known consequence of the Brauer-Hasse- Noether-Albert Theorem.

Theorem 1.4.3 Let F be a number field and denote by Pl(F ) the set of all primes (finite and infinite) of F . Let A be a central simple F -algebra having local Schur indices (mp(A))p∈Pl(F ). Then ind(A) = lcm(mp(A): p ∈ Pl(F )).

If a local Schur index equals 1, then the only possible local Hasse invariant 1 is 1 . If a local Schur index is 2, then the only possible local Hasse invariant 1 is 2 . Therefore, for a fixed number field F , division rings with center F and Schur index at most 2, are uniquely determined up to isomorphism by their local Schur indices. This is no longer true for division rings with bigger Schur index. Let L be a number field containing F and let A be a central simple F -algebra. Here, it is clear that ind(L ⊗F A) divides ind(A). One says that L splits A if and only if L ⊗F A is a matrix ring over L, if and only if ind(L ⊗F A) = 1.

6 1.4 number fields

Remark that if A is a division algebra of degree a prime number, then L does not split A if and only if L ⊗F A is a division algebra. Let L/F be number fields, R be the ring of integers of F , S the ring of integers of L and P a finite prime of S. One finds that P ∩ R = p, a prime in R, and one says that P lies over p. The primes lying over a given prime p are e1 es the P1,..., Ps which occur in the prime decomposition of pS = P1 ··· Ps . One also says that Pi divides p. The exponents ei are called the ramification Pi indices and are denoted by ep (L/F ). If P is a prime lying over p in the extension L/F , then R/p can be viewed as a subfield of S/P and one calls the degree of S/P over R/p the residue degree of P over p and denote it by P fp (L/F ). If L/F is a normal extension and p is a prime of R, then the Galois group Gal(L/F ) permutes the primes lying over p transitively. Therefore, both the ramification index and residue degree are independent on the choice of primes lying over p and one writes ep(L/F ) and fp(L/F ). One also knows the following multiplicative rules for normal extensions, see [Wei98, Proposition 1.6.2].

Theorem 1.4.4 Let K/L/F be normal extensions of number ﬁelds. Let p be a prime in F and P a prime in L lying over p. Then

1. ep(K/F ) = eP(K/L)ep(L/F );

2. fp(K/F ) = fP(K/L)fp(L/F ).

Because of these rules, the notations ep(L/F ) and fp(L/F ) are unambiguous for normal extensions L/F/Q and rational primes p. Similar as for ramiﬁcation index and residue degree, one can also restrict the computation of local Schur indices to rational primes when working over abelian number ﬁelds. The following theorem is due to Mark Benard [Ben72].

Theorem 1.4.5 (Benard) Let F be an abelian number field and A a central simple F -algebra. As p runs over the set of primes lying over the same (infinite or finite) rational prime p, the local indices mp(A) are all equal to the same positive integer.

One calls the integer deﬁned by the previous theorem the p-local index of A and denote it by mp(A).

7 preliminaries

The following proposition follows from [Wei98, Proposition 1.7.5 and Theo- rem 2.3.2].

Proposition 1.4.6 Let K/F be a normal extension of number ﬁelds, p a prime number and let Ffp and K›p be completions of respectively F and K with respect to p. Then ep(K›p/Ffp) = ep(K/F ) and fp(K›p/Ffp) = fp(K/F ). Furthermore, ep(K›p/Ffp)fp(K›p/Ffp) = [K›p : Ffp].

The next proposition follows from [Wei98, Corollary 2.5.9].

Proposition 1.4.7 Let F be a ﬁnite normal extension of Q and R be the ring of integers of F . Let p be a prime number. If r is the number of prime ideals in which pR splits, then ep(F/Q)fp(F/Q)r = [F : Q].

The following theorem can be used to decide if some ﬁeld splits a division ring and can be found in [Rei75, Corollary 31.10] and [Deu68, Satz 2 on p. 118].

Theorem 1.4.8 Let K be an abelian number ﬁeld extending the number ﬁeld F . Let D be a division ring with center F and let p be a rational prime. Then mp(K ⊗F D) = 1 if and only if mp(D) | [K›p : Ffp].

We conclude this section with some computations and easy examples that we need later. One denotes by φ Euler’s totient function. For cyclotomic ﬁelds, the ramiﬁcation index and residue degree is easy to compute, as shown in [Wei98, Theorems 7.2.4 and 7.4.3].

Theorem 1.4.9 s 0 Let Q(ζm) be a cyclotomic ﬁeld and p a prime number. Write m = p m 0 s with gcd(p, m ) = 1. One calculates that ep(Q(ζm)/Q) = φ(p ) and fp(Q(ζm)/Q) = om0 (p). √ Example 1.4.10 An easy computation shows that e2(Q√( 2)/Q) = 2. By Proposition 1.4.7, it now automatically follows that f2(Q( 2)/Q) = 1.

8 1.4 number fields

Example 1.4.11 Consider the following ﬁelds for a prime number p 6= 2:

Q(ζp)

−1 Q(ζp + ζp )

Q Since

ep(Q(ζp)/Q) = p − 1 = [Q(ζp): Q] −1 −1 = [Q(ζp): Q(ζp + ζp )][Q(ζp + ζp ): Q] and −1 −1 ep(Q(ζp)/Q) = ep(Q(ζp)/Q(ζp + ζp ))ep(Q(ζp + ζp )/Q), it follows that −1 −1 ep(Q(ζp)/Q(ζp + ζp )) = [Q(ζp): Q(ζp + ζp )] = 2 and p − 1 e ( (ζ + ζ−1)/ ) = [ (ζ + ζ−1): ] = . p Q p p Q Q p p Q 2 −1 Therefore, it immediately follows that fp(Q(ζp)/Q) = fp(Q(ζp)/Q(ζp+ζp )) = −1 fp(Q(ζp + ζp )/Q) = 1. The following example can be found in [Lam05, Theorems 6.2.5 and 6.2.24].

Ä ä Example 1.4.12 Let A = −1,−1 . Then ⊗ A = −1,−1 has Schur index Q R Q R 2, so m∞(A) = 2. One can also show that m2(A) = 2 and mp(A) = 1 for all odd prime numbers p. Fixed subﬁelds under cyclic groups are easy to compute in extensions of Q by a primitive root of unity of prime order, see [DF04, Example 14.5.2] for a proof. Lemma 1.4.13 Let p be a prime number and r be an integer coprime to p. Let G denote r the cyclic group generated by the automorphism Q(ζp) → Q(ζp): ζp 7→ ζp . G r rop(r)−1 Then Q(ζp) = Q(ζp + ζp + ... + ζp ).

9 preliminaries

1.5 crossed products

We recall some basic facts about crossed products, based on the books of Donald S. Passman [Pas89] and Irving Reiner [Rei75]. α If R is an associative ring and G is a group, then R ∗τ G denotes the crossed α product with action α : G → Aut(R) and twisting τ : G×G → U(R), i.e. R∗τ G L is the associative ring g∈G Rug with multiplication given by the following rules: uga = αg(a)ug and uguh = τ(g, h)ugh, for a ∈ R and g, h ∈ G. Recall that a classical crossed product, notated by (L/F, τ), is a crossed product α L ∗τ G, where L/F is a finite Galois extension, G = Gal(L/F ) is the Galois group of L/F and α is the natural action of G on L. Let F be a field and ζ be a root of unity in an extension of F . If Gal(F (ζ)/F ) = hσni is cyclic of α order n, one can consider the cyclic cyclotomic algebra F (ζ) ∗τ Gal(F (ζ)/F ), m where α is the natural action on F (ζ), i.e. α m = σ . Also, τ(g, h) is a root σn n of unity for every g and h in Gal(F (ζ)/F ) and τ is completely determined by un = ζc. One denotes this cyclic cyclotomic algebra by (F (ζ)/F, σ , ζc). σn n A classical crossed product (L/F, τ) is always a central simple F -algebra, see [Rei75, Theorem 29.6] for a proof. Theorem 1.5.1 Let L/F be a finite Galois extension and G = Gal(L/F ) its Galois group. For any twisting τ : G × G → U(L), the classical crossed product (L/F, τ) is a central simple F -algebra.

If the twisting τ is cohomologically trivial, then the classical crossed product is isomorphic to a matrix algebra over its center. Denote the matrix associated to an endomorphism f in a basis B as [f]B. The following theorem from [Rei75, Corollary 29.8] constructs an explicit isomorphism when τ = 1. Theorem 1.5.2 Let L/F be a ﬁnite Galois extension and n = [L : F ]. The classical crossed product (L/F, 1) is isomorphic (as F -algebra) to Mn(F ). Moreover, an isomorphism is given by

ψ :(L/F, 1) → EndF (L) → Mn(F ) 0 0 xuσ 7→ x ◦ σ 7→ [x ◦ σ]B,

for x ∈ L, σ ∈ Gal(L/F ), B an F -basis of L and where x0 denotes multiplication by x on L.

10 1.6 group rings

The following theorem is due to Gabriela Olteanu [Olt09, Proposition 2.13].

Theorem 1.5.3 (Olteanu)

Let A = (F (ζn)/F, σ, ζm) for F a number field. If p is a prime of F , then mp(A) divides m. If mp(A) 6= 1 for a finite prime p, then p divides n in F . Allen Herman gave a nice review on the computation of local indices for cyclic cyclotomic algebras in [Her] and he implemented this into the GAP- package wedderga [BHK+14]. Lemma 1.5.4 (Herman) c A cyclic cyclotomic algebra A = (F (ζn)/F, σ, ζn) over an abelian number field F has local index 2 at an infinite prime if and only if F ⊆ R, n > 2 c and ζn = −1. The following lemma originates from Gerald Janusz [Jan75, Lemma 3.1]. Lemma 1.5.5 (Janusz ) Let p be an odd prime number and F an abelian number field. Let e = ep(F (ζn)/F ) and f = fp(F/Q). Then

c mp(F (ζn)/F, σ, ζn) ® pf − 1 n ´ = min l ∈ : ≡ 0 mod . N gcd(pf − 1, e) gcd(n, cl)

1.6 group rings

We recall some notions from the commendable review book of César Polcino Milies and Sudarshan K. Sehgal [PMS02]. Let G be a group (not necessarily finite) and R a unitary ring. One denotes by RG the set of all formal linear combinations of the form X α = rgg, g∈G where rg ∈ R and only a finite number of coefficients rg are different from 0. P P Given two elements α = g∈G rgg and β = g∈G sgg ∈ RG, one has that

11 preliminaries

α = β if and only if rg = sg for every g ∈ G. One turns the set RG into a ring via the following operations. The sum of two elements of RG is deﬁned component-wise: Ñ é Ñ é X X X rgg + sgg = (rg + sg)g. g∈G g∈G g∈G

Their product is deﬁned by Ñ é Ñ é X X X rgg · sgg = (rgsh)gh. g∈G g∈G g,h∈G

It is easy to verify that indeed, with the operations given, RG is a ring with unity. One can also deﬁne a product of elements in RG with elements λ ∈ R by X X λ · rgg = (λrg)g. g∈G g∈G

This way, RG turns into an R-module and when R is commutative, RG is even an algebra over R. The set RG, with the operations as above, is called the group ring of G over R. In the case where R is commutative, RG is also called the group algebra of G over R. When R = Z, one calls ZG the integral group ring, when R = Q, one calls QG the rational group algebra and when R = F is a finite field and G is a finite group, one says that FG is a finite group algebra. P Given an element α = g∈G rgg ∈ RG, one defines the support of α to be the set of elements that effectively appear in the expression of α, i.e.

supp(α) = {g ∈ G : rg 6= 0}.

Let R be a ring, G a ﬁnite group and N a normal subgroup of G such that |N| is invertible in R. It is easy to see that the map G → GN“ given by g 7→ gN“ is a group epimorphism with kernel N. Hence, the R-linear extension yields an isomorphism

R(G/N) ' RGN.“ (1)

12 1.6 group rings

An R-module M is called semisimple if every submodule of M is a direct summand. A ring R is called semisimple if the regular module RR is semisimple. Semisimple rings have a clear structure as shown in [PMS02, Theorem 2.5.7]. Theorem 1.6.1 Let R be a ring. Then R is semisimple if and only if R is a direct sum of a ﬁnite number of minimal left ideals. Heinrich Maschke provided necessary and suﬃcient conditions on R and G for the group ring RG to be semisimple, see [PMS02, Theorem 3.4.7]. Theorem 1.6.2 (Maschke) Let G be a group and R a ring. Then, the group ring RG is semisimple if and only if the following conditions hold:

1. R is a semisimple ring;

2. G is ﬁnite;

3. |G| is invertible in R.

Since fields are always semisimple, Maschke’s Theorem is even easier to state in the case when R is a field. Corollary 1.6.3 Let G be a finite group and let F be a field. Then, FG is semisimple if and only if the characteristic of F does not divide |G|.

The structure of semisimple abelian group rings is well studied by Sam Perlis and Gordon Loftis Walker in [PW50]. Theorem 1.6.4 (Perlis-Walker) Let G be a ﬁnite abelian group of order n and let F be a ﬁeld such that the characteristic of F does not divide n. Then M FG ' kdF (ζd), d|n

nd where kd = and nd is the number of elements of order d in G. [F (ζd):F ]

For F = Q, one gets the following corollary.

13 preliminaries

Corollary 1.6.5 Let G be a ﬁnite abelian group of order n. Then M QG ' kdQ(ζd), d|n

where kd is the number of cyclic subgroups of order d of G (or equivalently, kd is the number of cyclic factors of G of order d).

1.7 wedderburn-artin decomposition

An element e in a ring R is called an idempotent if e2 = e. Clearly, 0 and 1 are idempotents. An idempotent diﬀerent from these is said to be non-trivial. For a unitary ring R, a matrix unit in Mn(R) is a matrix whose entries are all 0 except in one cell, where it is 1. Clearly matrix units with a 1 on the diagonal, are idempotents in Mn(R). When R is a semisimple ring, the left ideals of R are determined by idempotents as shown in [PMS02, Theorem 2.5.10]. Theorem 1.7.1 Let R be a ring. Then R is semisimple if and only if every left ideal L of R is of the form L = Re, where e is an idempotent of R.

Therefore, one can use the idempotents to characterize the decomposition of semisimple rings as direct sums of minimal left ideals. We refer to [PMS02, Theorem 2.5.11] for a proof. Theorem 1.7.2 Lt Let R = i=1 Li be a decomposition of a semisimple ring as a direct sum of minimal left ideals. There exists a family {e1, . . . , et} of elements of R such that: each ei is a non-zero idempotent element, if i 6= j, then eiej = 0, 0 00 e1 + ··· + et = 1 and each ei can not be written as ei = ei + ei , where 0 00 0 00 0 00 ei, ei are idempotents such that ei, ei 6= 0 and eiei = 0, for all 1 ≤ i ≤ t. Conversely, if there exists a family of idempotents {e1, . . . , et} satisfying the previous conditions, then the left ideals Li = Rei are minimal and Lt R = i=1 Li. Such a set of idempotents is called a complete set of orthogonal primitive idempotents of the ring R. Note that such a set is not uniquely determined.

14 1.7 wedderburn-artin decomposition

Example 1.7.3 Consider the matrix ring M2(F ) over a ﬁeld F . The sets

ßÅ 1 0 ã Å 0 0 ã™ , 0 0 0 1

and ßÅ 1 1 ã Å 0 −1 ã™ , 0 0 0 1 are two diﬀerent complete sets of orthogonal primitive idempotents. Given a decomposition of a semisimple ring R as a direct sum of minimal left ideals, one can group isomorphic left ideals together. The sum of all left ideals isomorphic to one in the decomposition, turns out to be a minimal two-sided ideal of R, which is simple as a ring. Also, the decomposition of R as direct sums of two-sided ideals is related to a family of idempotents. (See [PMS02, Theorem 2.6.8].) Theorem 1.7.4 Ls Let R = i=1 Ai be a decomposition of a semisimple ring as a direct sum of minimal two-sided ideals. There exists a family {e1, . . . , es} of elements of R such that: each ei is a non-zero central idempotent element, if i 6= j, then eiej = 0, e1 + ··· + es = 1 and each ei can not be written 0 00 0 00 0 00 as ei = ei + ei , where ei, ei are central idempotents such that ei, ei 6= 0 0 00 and eiei = 0, 1 ≤ i ≤ s.

The elements {e1, . . . , es} are called the primitive central idempotents of R and they give rise to the well known Wedderburn-Artin Theorem. Theorem 1.7.5 (Wedderburn-Artin) A ring R is semisimple if and only if it is a direct sum of matrix rings over division rings.

From now on, G denotes an arbitrary ﬁnite group and F a ﬁeld such that FG is semisimple. By Maschke’s Theorem (Theorem 1.6.2) this is equivalent with saying that the order of G is coprime to the characteristic of F . The following example is given by Charles W. Curtis and Irving Reiner in [CR81, Example 7.39].

Example 1.7.6 Let D2n denote the dihedral group of order 2n, i.e.

n 2 −1 D2n = a, b : a = 1, b = 1, bab = a .

15 preliminaries

We have M QD2n ' Ad, d|n where ß Q ⊕ Q d = 1, 2 Ad = −1 M2(Q(ζd + ζd )) d 6= 1, 2. The classical method for computing primitive central idempotents in FG involves characters of the group G. All the characters of any finite group are assumed to be characters in F , a fixed algebraic closure of the field F . For χ(1) P −1 an irreducible character χ of G, e(χ) = |G| g∈G χ(g )g is the primitive central idempotent of FG associated to χ and eF (χ) is the only primitive central idempotent e of FG such that χ(e) 6= 0. The field of character values of χ over F is defined as F (χ) = F (χ(g): g ∈ G), that is, the field extension of F generated over F by the image of χ. The automorphism group Aut(F ) P P acts on FG by acting on the coefficients, that is, σ g∈G agg = g∈G σ(ag)g, for σ ∈ Aut(F ) and ag ∈ F . Following Toshihiko Yamada [Yam74], one knows that X eF (χ) = σe(χ). (2) σ∈Gal(F (χ)/F ) The known methods to compute the character table of a finite group are very time consuming. Therefore, in practical applications, the classical description of primitive central idempotents sometimes is of limited use. New methods for the computation of the primitive central idempotents in a group algebra do not involve characters. The main ingredient in this theory is the following element, introduced in [JLP03] by Eric Jespers, Guilherme Leal and Antonio Paques. If K ¢ H ≤ G, then let ε(H,K) be the element of QH ⊆ QG defined as ® K“ if H = K, ε(H,K) = Q M/K∈M(H/K)(K“ − Mc) if H 6= K, where M(H/K) denotes the set of minimal normal non-trivial subgroups of H/K. Furthermore, e(G, H, K) denotes the sum of the different G-conjugates of ε(H,K). Aurora Olivieri, Angel´ del R´ıoand Juan Jacobo Simónachieved a break- through in the study of primitive central idempotents associated to monomial characters in [OdRS04]. We will explain this in detail.

16 1.7 wedderburn-artin decomposition

Let H be a subgroup of G and χ a character of H. Then χG is given by

1 X χG(g) = χ◦(xgx−1), |H| x∈G where χ◦ is defined by χ◦(h) = χ(h), if h ∈ H, and χ◦(y) = 0 if y∈ / H. It is well known that χG is a character of G and it is called the induced character on G (see [Isa76, Corollary 5.3]). A (complex) character ψ of G is called monomial if there exists a subgroup H ≤ G and a linear character χ of H such that ψ = χG. The group G is called monomial if all its irreducible (complex) characters are monomial. Definition 1.7.7 A pair (H,K) of subgroups of G is called a Shoda pair if it satisfies the following conditions:

1. K ¢ H;

2. H/K is cyclic;

3. if g ∈ G and [H, g] ∩ H ⊆ K, then g ∈ H.

Now, one can rephrase an old theorem of Kenjiro Shoda [Sho33] as follows. Proposition 1.7.8 (Shoda) If χ is a linear character of a subgroup H of G with kernel K, then the induced character χG is irreducible if and only if (H,K) is a Shoda pair.

For monomial groups G, Shoda pairs are sufficient to be able to construct all primitive central idempotents of QG, see [OdRS04, Corollary 2.3]. Theorem 1.7.9 (Olivieri-del R´ıo-Simón) A finite group G is monomial if and only if every primitive central idempotent of QG is of the form αe(G, H, K), for α ∈ Q and (H,K) a Shoda pair of G.

So all primitive central idempotents of QG for monomial groups correspond to Shoda pairs of G. However, diﬀerent Shoda pairs can contribute to the same primitive central idempotent of G. This is expressed in [OdRS06, Proposition 1.4] in terms of a relation on Shoda pairs.

17 preliminaries

Proposition 1.7.10 (Olivieri-del R´ıo-Sim´on)

Let (H1,K1) and (H2,K2) be two Shoda pairs of a ﬁnite group G and consider α1, α2 ∈ Q such that ei = αie(G, Hi,Ki) is a primitive central idempotent of QG, for i = 1, 2. Then e1 = e2 if and only if a g ∈ G exists g g such that H1 ∩ K2 = K1 ∩ H2.

One says that a set S of Shoda pairs of G is a complete and non-redundant set of Shoda pairs of G if all primitive central idempotents of G can be realized by a pair in S and no two diﬀerent pairs in S determine the same idempotent in QG. Note that α is uniquely determined by the fact that αe(G, H, K) is an idempotent. If for example, the G-conjugates of ε(H,K) are orthogonal, then α = 1. This happens in the case when G is a strongly monomial group. For this one introduces strong Shoda pairs. Deﬁnition 1.7.11 A strong Shoda pair of G is a pair (H,K) of subgroups of G satisfying the following conditions:

1. K ≤ H ¢ NG(K);

2. H/K is cyclic and a maximal abelian subgroup of NG(K)/K;

g 3. for every g ∈ G \ NG(K), ε(H,K)ε(H,K) = 0.

A (complex) character ψ of G is said to be strongly monomial if there is a strong Shoda pair (H,K) of G and a linear character χ of H with kernel K such that ψ = χG. A group G is strongly monomial if all its irreducible (complex) characters are strongly monomial. The big advantage of working with strongly monomial groups is that one can describe the structure of the simple components of QG. The following results can be found in [OdRS04, Propositions 3.3 and 3.4, Theorem 4.7]. Proposition 1.7.12 (Olivieri-del R´ıo-Sim´on) The following are equivalent for a pair (H,K) of subgroups of G: 1. (H,K) is a strong Shoda pair of G;

2. (H,K) is a Shoda pair of G, H ¢ NG(K) and the G-conjugates of ε(H,K) are orthogonal.

18 1.7 wedderburn-artin decomposition

Moreover, if the previous conditions hold then CenG(ε(H,K)) = NG(K) and e(G, H, K) is a primitive central idempotent of QG.

Proposition 1.7.13 (Olivieri-del R´ıo-Sim´on) Let (H,K) be a strong Shoda pair of a ﬁnite group G and let k = [H : K], N = NG(K), n = [G : N], yK a generator of H/K and ψ : N/H → N/K a left inverse of the canonical projection N/K → N/H. The simple α algebra QGe(G, H, K) is isomorphic to Mn(Q(ζk)∗τ N/H) and the action and twisting are given by

i ψ(nH) i αnH (ζk) = ζk, if yK = y K and 0 j 0 −1 0 j τ(nH, n H) = ζk, if ψ(nn H) ψ(nH)ψ(n H) = y K,

for nH, n0H ∈ N/H and integers i and j.

Proposition 1.7.14 (Olivieri-del R´ıo-Sim´on) A ﬁnite group G is strongly monomial if and only if every primitive central idempotent of QG is of the form e(G, H, K), for (H,K) a strong Shoda pair of G.

In analogue to the Shoda pairs, one says that a set S of strong Shoda pairs of G is a complete and non-redundant set of strong Shoda pairs of G if all primitive central idempotents of G can be realized by a pair in S and no two diﬀerent pairs in S determine the same idempotent in QG. The following is proven in [OdR03, Proposition 2.4]. Proposition 1.7.15 (Olivieri-del R´ıo)

If (H,K) is a strong Shoda pair of a ﬁnite group G and N = NG(K), then N/H is isomorphic to a subgroup of the group of units of Z/[H : K]Z and in particular N/H is abelian.

Let k be a positive integer and Ck = hci a cyclic group of order k. Then there are isomorphisms

Gal(Q(ζk)/Q) → U(Z/kZ) → Aut(Ck) r r (ζk 7→ ζk ) 7→ r 7→ (σr : c → c ) Throughout this thesis we will abuse the notation and consider these isomorphisms as equalities. For example, a subgroup H of Aut(Ck) will be identiﬁed

19 preliminaries

H with a subgroup of U(Z/kZ) and with Gal(Q(ζk)/Q(ζk) ). In particular, for a strong Shoda pair (H,K) of G, with the notation of Proposition 1.7.13 and because H/K is a maximal abelian subgroup of N/K, the action α of the crossed α α product Q(ζk)∗τ N/H is faithful. Therefore, the crossed product Q(ζk)∗τ N/H can be described as a classical crossed product (Q(ζk)/F, τ), where F is the center of the algebra which is determined by the action of N/H on H/K. In this way, the Galois group Gal(Q(ζk)/F ) can be identified with N/H and with N/H this identification F = Q(ζk) . Examples of strongly monomial groups are abelian-by-supersolvable groups [OdRS04, Corollary 4.6]. All monomial groups of order smaller than 1000 are strongly monomial and the smallest monomial non-strongly monomial group is a group of order 1000, number 86 in the library of the GAP system [Olt07, GAP14]. Metabelian groups are also strongly monomial and provide an easier description for the strong Shoda pairs. Theorem 1.7.16 (Olivieri-del R´ıo-Simón) Let G be a finite metabelian group and let A be a maximal abelian subgroup of G containing the commutator subgroup G0. The primitive central idempotents of QG are the elements of the form e(G, H, K), where (H,K) is a pair of subgroups of G satisfying the following conditions:

1. H is a maximal element in the set

{B ≤ G : A ≤ B and B0 ≤ K ≤ B};

2. H/K is cyclic.

Theorem 1.7.16 and Proposition 1.7.13 allow one to easily compute the primitive central idempotents and the Wedderburn components of the rational group algebra of a ﬁnite metacyclic group. Recall that this is a group G having a normal cyclic subgroup N = hai such that G/N = hbNi is cyclic. Every ﬁnite metacyclic group G has a presentation of the form

G = a, b : am = 1, bn = at, ab = ar , where m, n, t, r are integers satisfying the conditions rn ≡ 1 mod m and m divides t(r − 1). Deﬁne σ ∈ Aut(hai) as σ(a) = ab = ar. When u is the order d of σ, it follows that u | n. For every d | u, let Gd = ha, b i.

20 1.7 wedderburn-artin decomposition

Lemma 1.7.17 With notations as above, the primitive central idempotents of QG are the elements of the form e(G, Gd,K) where d is a divisor of u and K is a subgroup of Gd satisfying the following conditions:

x 1. d = min{x | u : ar −1 ∈ K};

2. Gd/K is cyclic.

We apply this result to the speciﬁc case of split metacyclic groups with trivial kernel. Corollary 1.7.18 Let p and q be diﬀerent prime numbers, m and n positive integers and m n G = haio1hbi, with |a| = q and |b| = p . A complete and non-redundant set of strong Shoda pairs of G consists of two types:

Ä ¨ pi ∂ä 1. G, Li := a, b , i = 0, . . . , n;

Ä ¨ qj ∂ä 2. hai ,Kj := a , j = 1, . . . , m.

The simple components of QG are: (i) QGε (G, Li) ' Q ζpi , i = 0, . . . , n;

hbi (ii) QGε (hai ,Kj) ' Q ζqj ∗ hbi ' Mpn Q(ζqj ) , j = 1, . . . , m.

Proof From the notation haio1 hbi, it automatically follows that Cenhbi(a) = 1, or equivalently, that hbi acts faithfully on hai. Let σ be the automorphism of hai given by σ(a) = ab and assume that σ(a) = ar, with r ∈ Z. As the Ä¨ m−1 ∂ä kernel of the restriction map Aut(hai) → Aut aq has order qm−1, it ¨ m−1 ∂ intersects hσi trivially and therefore the restriction of σ to aq also has order pn. This implies that q ≡ 1 mod pn and thus q is odd. Therefore, Ä¨ j ∂ä Aut aq (= U(Z/qjZ)) is cyclic for every j = 0, 1, . . . , m and hσi is the unique subgroup of Aut(hai) of order pn. Hence, for every i = 1, . . . , m, the image of r in Z/qiZ generates the unique subgroup of U(Z/qiZ) of order pn. n j In particular rp ≡ 1 mod qm and rp 6≡ 1 mod q for every j = 0, . . . , n − 1.

21 preliminaries

Therefore, r 6≡ 1 mod q and hence G0 = ar−1 = hai. Using the description of strong Shoda pairs of G as given in Lemma 1.7.17 and the description of the associated simple algebra given in Proposition 1.7.13 and Theorem 1.5.2, we obtain the result. Although the description of the Wedderburn decomposition and the primitive central idempotents of QG is known for strongly monomial groups G, it is not trivial to obtain a description of a complete set of (non-central) orthogonal primitive idempotents. However, Eric Jespers, Gabriela Olteanu and Angel´ del R´ıoobtained such a description in [JOdR12, Theorem 4.5] for nilpotent groups. Q Let G be a ﬁnite nilpotent group, one writes G = p Sp, i.e. the direct product of its Sylow p-subgroups Sp for p dividing |G|. The 2-part of G is the 0 Q Sylow 2-subgroup S2. The 2 -part is the direct product p6=2 Sp of all Sylow p-subgroups for p 6= 2. Theorem 1.7.19 (Jespers-Olteanu-del R´ıo) Let G be a ﬁnite nilpotent group and (H,K) a strong Shoda pair of G. Set e = e(G, H, K), ε = ε(H,K), H/K = hai, N = NG(K) and let N2/K and H2/K = ha2i (respectively N20 /K and H20 /K = ha20 i) denote the 2-parts (respectively 20-parts) of N/K and H/K respectively. The group ha20 i has a cyclic complement hb20 i in N20 /K. A complete set of orthogonal primitive idempotents of QGe consists of the conjugates of bc20 β2ε by the elements of T20 T2TG/N , where T20 = 2 [N20 :H20 ]−1 {1, a20 , a20 , . . . , a20 }, TG/N denotes a right transversal of N in G and β2 and T2 are given according to the cases below.

1. If H2/K has a complement M2/K in N2/K, then β2 = M”2. More- over, if M2/K is cyclic, there exists b2 ∈ N2 such that N2/K is given by the following presentation

n k 2 2 b2 r ha2, b2 : a2 = b2 = 1, a2 = a2 i,

and if M2/K is not cyclic, there exist b2, c2 ∈ N2 such that N2/K is given by the following presentation

n k 2 2 2 b2 r ha2, b2, c2 : a2 = b2 = 1, c2 = 1, a2 = a2 ,

c2 −1 a2 = a2 , [b2, c2] = 1i,

2n−2 with r ≡ 1 mod 4 (or equivalently a2 is central in N2/K).

22 1.7 wedderburn-artin decomposition

2 2k−1 2n−2 a) T2 = {1, a2, a2, . . . , a2 }, if a2 is central in N2/K (unless n ≤ 1) and M2/K is cyclic; and b) otherwise

n−2 n−2 2 [N2:H2]/2−1 2 2 +1 T2 = {1, a2, a2, . . . , a2 , a2 , a2 ,..., n−2 2 +[N2:H2]/2−1 a2 }.

2. If H2/K does not have a complement in N2/K, there exist elements b2, c2 ∈ N2, such that N2/K is given by the following presentation

n k n−1 2 2 2 2 b2 r ha2, b2, c2 : a2 = b2 = 1, c2 = a2 , a2 = a2 ,

c2 −1 a2 = a2 , [b2, c2] = 1i,

with r ≡ 1 mod 4 and we set m = [H20 : K]/[N20 : H20 ]. 2 2k−1 a) β2 = b“2 and T2 = {1, a2, a2, . . . , a2 }, if either m = 1 or the order of 2 modulo m is odd and n − k ≤ 2 and

2n−2 2n−2 1+xa2 +ya2 c2 b) β2 = b“2 2 and

2 2k−1 2 2k−1 T2 = {1, a2, a2, . . . , a2 , c2, c2a2, c2a2, . . . , c2a2 }

with k k Ä [N20 :H20 ] 2 −2 ä x, y ∈ Q a20 , a2 + a2 , satisfying (1 + x2 + y2)ε = 0, if m 6= 1 and either the order of 2 modulo m is even or n − k > 2.

For rational group algebras of arbitrary ﬁnite metacyclic groups, we would like to be able to give a similar description of a complete set of orthogonal primitive idempotents as in the previous theorem. Unfortunately, the approach of Theorem 1.7.19 does not apply here. To show this, the following example was given in [JOdR12, Remark 4.8].

−1 2 Example 1.7.20 Let G = C7 o1 C3 = hai o1 hbi, with b ab = a and e = e(G, hai, 1) = ε(hai, 1). One can check that there does not exist a complete√ set of orthogonal primitive idempotents of QGe = Q(ζ7)∗hbi = (Q(ζ7)/Q( −7), 1) formed by Q(a)-conjugates of bbε.

23 preliminaries

Nevertheless, in Chapter 2, and more particular in Corollary 2.2.5, we show how to overcome the diﬃculties and we will produce a complete set of orthogonal primitive idempotents for Q(C7 o1 C3).

1.8 Z-orders

We collect some basic results about Z-orders from the book of Sudarshan Sehgal [Seh93]. A subring O of a finite dimensional Q-algebra A is called a Z-order (or order) if it is a finitely generated Z-module such that QO = A. For example, the ring of integers R of a number field F is an order in F . If G is a finite group, then ZG is an order in QG and Z (ZG) is an order in Z (QG). More generally, RG is an order in FG. If O is an order in a division ring D, then Mn (O) is an order in Mn (D). An order is said to be maximal if it is not properly contained in a bigger order, within the same finite dimensional Q-algebra. The following lemmas provide examples of maximal orders and can be found in [Rei75, Theorem 8.6] and [Seh93, Lemmas 4.4 and 4.5]. Lemma 1.8.1 The ring of integers of a number field F is the unique maximal order in F .

Lemma 1.8.2 Let A be a ﬁnite dimensional Q-algebra with Wedderburn decomposition L i Aei. Let O be a maximal order in A, then L 1. O = i Oei. Moreover, Oei is a maximal order of Aei;

2. Mn(O) is a maximal order in Mn(A).

Lemma 1.8.3 Every order in A is contained in a maximal order.

L Example 1.8.4 Let G be a ﬁnite abelian group and QG = d Q(ζd) with L possible repetition. Then d Z[ζd] is the unique maximal order in QG and L hence ZG ⊆ d Z[ζd].

24 1.8 Z-orders

Let R be a ring. The unit group of R is denoted U(R). The following lemmas can be found in [Seh93, Lemmas 4.2 and 4.6] Lemma 1.8.5 The intersection of two orders in A is again an order in A.

Lemma 1.8.6

Suppose O1 ⊆ O2 are two orders in A. The index of their unit groups [U(O2): U(O1)] is ﬁnite. Furthermore, if u ∈ O1 is invertible in O2, then −1 u ∈ O1.

Therefore, for two arbitrary orders O1, O2 in A, we have that the index [U(O2): U(O1 ∩ O2)] is finite, in other words U(O1) and U(O2) are commensurable. The following theorem on orders follows from deep results by Carl Ludwig Siegel in [Sie43] and Armand Borel and Harish-Chandra in [BHC62]. Theorem 1.8.7 (Siegel-Borel-Harish-Chandra) The group of units of an order of a finite dimensional Q-algebra is finitely generated.

It follows that the unit groups U(ZG) and Z(U(ZG)) = U(Z(ZG)) are finitely generated if G is a finite group. Let Γ be a finitely generated abelian group. The rank of all free abelian subgroups of finite index in Γ is an invariant of the group, called the rank of Γ. Note that replacing generators of Γ by powers of themselves yields generators of a subgroup of finite index. We will use this fact throughout this document without explicitly mentioning it. As an immediate application of Theorem 1.8.7, one obtains that U(R) is finitely generated, when R is the ring of integers in a number field. One can even determine the rank of U(R). This is one of the most famous results on units of orders. Theorem 1.8.8 (Dirichlet’s Unit Theorem) Let F be a number field of degree n = r + 2s over Q, where r is the number of real embeddings of F and s is the number of pairs of complex embeddings of F . Let R be the ring of integers of F . The unit group U(R) is a finitely generated abelian group. Moreover, U(R) = C × A, where C is a finite cyclic group and A is torsion free of rank r + s − 1.

25 preliminaries

1.9 congruence subgroup problem

Let K be a number field and A a finite dimensional semisimple K-algebra. Let F be a splitting field of A over K and fix an isomorphism h : F ⊗K A →

Mn1 (F ) × · · · × Mnm (F ) of F -algebras. Let hi : F ⊗K A → Mni (F ) be the composition of h with the projection on the i-th component. By deﬁnition, Qm the reduced norm of a ∈ A is given by nrdA/K (a) = i=1 det(hi(1 ⊗ a)). Examples 1.9.1 Let K be a number ﬁeld.

1. If Gal(K/Q) = {σ1, . . . , σn}, then one has the following isomorphism

n K ⊗Q K → K : x ⊗ y 7→ (xσi(y))1≤i≤n. Qn Therefore nrdK/Q(x) = i=1 σi(x).

Ä a,b ä 2. If A is the quaternion algebra K with splitting ﬁeld E, then the following is an isomorphism of E-algebras:

E ⊗K A → M2(E) Ç √ √ √ å √ √ √ x + y a z b + t ab 1 ⊗ (x + y a + z b + t ab) 7→ √ √ √ z b − t ab x − y a √ √ √ 2 2 2 2 and nrdA/K (x + y a + z b + t ab) = x − ay − bz + abt . In particular, let K be a number field and D a division algebra which is finite dimensional over K with degree m. If A = Mn(D) and O is an order in D, then, one can prove that nrdA/K (A) ⊆ K and nrdA/K (Mn(O)) ⊆ Z(O) (see [CR81, Corollary 26.2]). Also, a ∈ A is a unit in A if and only if nrdA/K (a) is invertible (see [CR81, page 170]). One denotes by GLn(O) the invertible matrices in Mn(O), and by SLn(O) the matrices of reduced norm 1 in GLn(O). nm It is easy to verify that nrdA/K (Z(GLn(O))) = U(Z(O)) . By Dirichlet’s Unit Theorem 1.8.8, U(Z(O)) is a finitely generated abelian group, hence nm [GLn(O): Z(GLn(O))SLn(O)] ≤ [U(Z(O)) : U(Z(O)) ] is finite. Let m be a positive integer, one calls

SLn(O, m) = {M ∈ SLn(O): M − In ∈ Mn(mO)}

the congruence subgroup of level m. Clearly, SLn(O, m) has ﬁnite index in SLn(O). The Congruence Subgroup Problem (CSP) asks for the converse of this statement:

26 1.9 congruence subgroup problem

Does every subgroup of SLn(O) of ﬁnite index contain a congruence subgroup?

For a more detailed survey on the Congruence Subgroup Problem, see for example [Sur03, PR10]. More relevant for our investigations, is the following problem. Let I be a non-zero ideal of O. Matrices having 1 at every entry on the diagonal and only one oﬀ-diagonal entry in I and 0 in all other oﬀ-diagonal entries, are called elementary matrices modulo I. The group generated by these elementary matrices is written as En(I). A relevant question is:

Is En(I) of ﬁnite index in SLn(O)?

This problem seems to be strongly related to the Congruence Subgroup Prob- lem. When the CSP has an “almost” positive answer, it is proven that indeed En(I) is of ﬁnite index in SLn(O). The explanation of what an “almost” positive answer to CSP is, is out of the scope of this thesis. Again, the interested reader is referred to [Sur03, PR10]. Leonid N. Vaserˇste˘ınand, independently Jacques Tits, have proven a positive answer to the question for n ≥ 3 [Vas73, Tit76] and Tyakal Nanjundiah Venkataramana found a proof for the case n = 2 [Ven94]. Parts of these results were proven earlier by others as well, for example by Hyman Bass [Bas64], Leonid N. Vaserˇste˘ın[Vas72] and Bernhard Liehl [Lie81]. Actually, they have proven the statements in the setting of linear algebraic groups, but we state it only in the context essential to our investigations.

Theorem 1.9.2 (Bass-Vaserˇste˘ın-Tits) Let O be an order in a ﬁnite dimensional rational division algebra. If n ≥ 3, then [SLn(O): En(I)] < ∞ for every non-zero ideal I of O.

Theorem 1.9.3 (Vaserˇste˘ın-Liehl-Venkataramana) Let O be an order in a finite dimensional rational division algebra D. If D is different from Q, a quadratic imaginary extension of Q and a totally definite quaternion algebra with center Q, then [SL2(O): E2(I)] < ∞ for all non-zero ideals I of O.

For the case n = 1, almost nothing is known, except for a result of Ernst Kleinert [Kle00a].

27 preliminaries

Theorem 1.9.4 (Kleinert)

When O is an order in a non-commutative division ring D, SL1(O) is ﬁnite if and only if D is a totally deﬁnite quaternion algebra.

The above theorems lead us to the deﬁnition of exceptional simple algebras.

Deﬁnition 1.9.5 A simple ﬁnite dimensional rational algebra is an exceptional simple algebra if it is of one of the following types:

EC1: a non-commutative division ring other than a totally deﬁnite quaternion algebra;

EC2:a 2 × 2-matrix ring over the rationals, a quadratic imaginary extension of the rationals or over a totally deﬁnite quaternion algebra over Q.

Let G be a finite group and F a number field. We say that FG contains an exceptional component if one of the simple algebras in the Wedderburn decomposition of FG is an exceptional simple algebra. In joined work with Florian Eisele and Ann Kiefer, we showed that the exceptional components of a group algebra FG over a number field F are of a very restricted type, [EKVG15, Theorems 3.1 and 3.5, Proposition 3.3].

Theorem 1.9.6 (Eisele-Kiefer-Van Gelder)

If G is a ﬁnite subgroup of GL2(F ), for F a quadratic imaginary extension of the rationals, such that G spans M2(F ) over F , then G is solvable, |G| = 2a3b for some a, b ∈ N and F is one of the following ﬁelds: √ 1. Q( −1); √ 2. Q( −2); √ 3. Q( −3).

Furthermore, elements of a ﬁnite subgroup G of GL2(Q) can only have prime power orders 1, 2, 3 and 4.

The following proposition turns the classification of all finite subgroups of √ GL2(F ) for F ∈ Q( −d): d = 0, 1, 2, 3 into a finite problem.

28 1.9 congruence subgroup problem

Proposition 1.9.7 (Eisele-Kiefer-Van Gelder) √ Let F ∈ Q( −d): d = 0, 1, 2, 3 , and let G ≤ GL2(F ) be a ﬁnite group. The group G can be embedded in the ﬁnite group GL(2, 25).

Theorem 1.9.8 (Eisele-Kiefer-Van Gelder)

Let G be a ﬁnite group. If G is a subgroup of GL2(D), for D a totally deﬁnite quaternion algebra with center Q, such that G spans M2(D) over Q, then D is one of the following algebras: Ä ä 1. −1,−1 ; Q Ä ä 2. −1,−3 ; Q Ä ä 3. −2,−5 . Q

Due to Theorems 1.9.6 and 1.9.8, the exceptional simple algebras appearing in a group algebra restrict to the following. Corollary 1.9.9 If a simple ﬁnite dimensional rational algebra is an exceptional component of some group algebra FG for some number ﬁeld F , then it is of one of the following types:

EC1: a non-commutative division ring other than a totally deﬁnite quaternion algebra;

√ √ √ Ä −1,−1 ä EC2: M2( ), M2( ( −1)), M2( ( −2)), M2( ( −3)), M2 , Q Q Q Q Q Ä −1,−3 ä Ä −2,−5 ä M2 , M2 . Q Q

Note that, Theorems 1.9.6 and 1.9.8 could also have been reduced from clas- sifications of Behnam Banieqbal [Ban88] and Gabriele Nebe [Neb98, Theorems 6.1 and 12.1], however their proofs are very long and tedious. Therefore, we provided elementary proofs in [EKVG15] for the very specific cases were we want to use Banieqbal’s or Nebe’s results. Let A be a finite dimensional rational simple algebra and denote the absolute value of the reduced norm nrdA/Q : A → Q by NA/Q : A → Q≥0. For any order O of A, we have NA/Q(O) ⊆ N ∪ {0}.

29 preliminaries

Examples 1.9.10 √ 1. Let d be a square-free positive integers, then N √ (x + y −d) = √ √ Q( −d)/Q x2 + dy2 for each x + y −d ∈ Q( −d).

2. Let a, b be negative rational numbers√ and√ let A be the quaternion algebra Ä a,b ä √ 2 2 2 2 , then NA/ (x + y a + z b + t ab) = x − ay − bz + abt for Q √ Q√ √ each x + y a + z b + t ab ∈ A.

One says that an order O in A is left norm Euclidean if for every x, y ∈ O, with y 6= 0, there exist q, r ∈ O such that x = qy + r with N(r) < N(y). It is easy to show that a left norm Euclidean order necessarily is a maximal order, for a proof see for example [CCL13, Proposition 2.8]. Theorems 1.9.6 and 1.9.8 show that if M2(D) is√ an exceptional component of a group algebra FG, then D is either a field Q( −d), with d ∈ {0, 1, 2, 3}, Ä ä or a quaternion algebra a,b , with (a, b) ∈ {(−1, −1), (−1, −3), (−2, −5)}. It Q is well known that in the 4 commutative cases, the ring of integers O of D is a Euclidean domain (see for example [Wei98, Proposition 6.4.1]), and actually the reduced norm turns O into a (left) norm Euclidean order. Moreover, the listed quaternion algebras are the only possible totally definite quaternion algebras over Q having a left norm Euclidean order O and O is the unique maximal order (see [Fit12, Theorem 2.1] or [CCL13, Theorem 1.6]). Corollary 1.9.11 (Eisele-Kiefer-Van Gelder) Let G be a finite group and F a number field with the property that FG contains an exceptional component M2(D). Then D contains a left norm Euclidean order O, which is necessarily the unique maximal order of D. The corresponding maximal orders are listed in Table 1 below.

Table 1: Maximal orders Division ring Maximal order

QZ√ √ Q(√−1) Z[√−1] ( −2) [ −2] Q Z √ √ î 1+ −3 ó Q( −3) Z 2 continued

30 1.10 finite subgroups of exceptional simple algebras

Division ring Maximal order Ä ä −1,−1 [1, i, j, 1 (1 + i + j + ij)] Q Z 2 Ä −1,−3 ä 1 1 Z[1, i, 2 (1 + j), 2 (i + ij)] Ä Q ä −2,−5 [1, 1 (2 + i − ij), 1 (2 + 3i + ij), 1 (1 + i + j)] Q Z 4 4 2

1.10 finite subgroups of exceptional simple algebras

We recall the classification of finite subgroups of division rings from Shimshon Avraham Amitsur [Ami55]. This classification splits into 2 parts: a list of some Z-groups and a list of non-Z-groups. Recall that Z-groups are groups for which all Sylow p-subgroups are cyclic. We use notations from [SW86, Theorems 2.1.4, 2.1.5] and [CdR14, Theorem 2.2]. Theorem 1.10.1 (Amitsur)

(Z) The Z-groups which are ﬁnite subgroups of division rings are: a) the ﬁnite cyclic groups;

b) Cm o2 C4, with m odd and C4 acting by inversion on Cm;

c) Cm ok Cn, with n 6= 1, gcd(m, n) = 1 and, using the following notation

Pp = Sylow p-subgroup of Cm;

Qp = Sylow p-subgroup of Cn;

Xp = {q | n : q prime and [Pp,Qq] 6= 1}; Y Rp = Qq;

q∈Xp Q we assume that Cn = p|m Rp and the following properties hold for every prime number p | m and q ∈ Xp:

i. q · o vq (k) (p) o mn (p); q - |Pp| |Rp|

ii. if q is odd or p ≡ 1 mod 4, then vq(p − 1) ≤ vq(k);

iii. if q = 2 and p ≡ −1 mod 4, then v2(k) is either 1 or greater than v2(p + 1).

31 preliminaries

(NZ) The ﬁnite subgroups of division rings which are not Z-groups are: a) O∗ = s, t :(st)2 = s3 = t4 (binary octahedral group); b) SL(2, 5);

c) Q4k with k even; d) SL(2, 3) × M, with M a group in (Z) of order coprime to 6 and o|M|(2) odd;

e) Q8 × M, with M a group in (Z) of odd order such that o|M|(2) is odd.

Remark 1.10.2 Assume that G = Cm ok Cn is a group satisfying the hypothesis in (Z)(c) in Theorem 1.10.1. Since G is not cyclic, Cn acts non-trivially. Let p be a prime divisor of m, such that Cn acts non-trivially on the Sylow p-subgroup Pp of Cm. Let q1, . . . , qh be the prime divisors of n such that for

each qi, Qqi acts non-trivially on Pp, so Rp = Qq1 ··· Qqh . Let kp be the order

of the kernel of the action of Rp on Pp. Then Pp okp Rp is a direct factor of G. From the conditions on (Z)(c) it follows that for every prime divisor qi of Q |Rp|, we have vqi (p − 1) ≤ vqi (k). Since Cn = p|m Rp is a direct product, all Q Xp are mutually disjoint, k = p|m kp and kp is only divisible by q if q ∈ Xp.

Therefore vqi (k) = vqi (kp). γ Let p be the order of Pp, and let the action of Rp on Pp be deﬁned by |Rp| σ : Rp → Aut(Pp), then Rp/Ker(σ) ' Im(σ). It follows that divides kp γ−1 |Rp| |Aut(Pp)| = p (p − 1). Thus, divides p − 1. kp βi αi On the other hand, for all 1 ≤ i ≤ h, let qi and qi be the order of Qqi

and the order of the kernel of the action of Qqi on Pp, respectively. Hence, β1 β α α h 1 h α |Rp| = q ··· q and kp = q ··· q . Given a ﬁxed i, as Pp i Qqi is not 1 h 1 h oqi |Rp| cyclic, βi 6= αi, so that qi divides . It now follows that kp Å ã |Rp| 1 ≤ vqi ≤ vqi (p − 1) ≤ vqi (kp). (3) kp

Note that ones deduce from (3) that every prime divisor of |Rp| is a prime |Rp| divisor of both and kp. In particular, this means that any prime divisor kp n of n divides both k and k . Finally, by (3) every αi is not zero, this implies

that no Qqi acts faithfully on Pp, and neither does Rp. Hence kp 6= 1, and in particular k 6= 1.

32 1.11 cyclotomic units

Gabriele Nebe gave a list of maximal finite subgroups of GLn(D), spanning Mn(D) over Q, for D a totally definite quaternion algebra with center of degree d over Q, such that nd ≤ 10. We recall this result for the case n = 2 and d = 1, see [Neb98, Theorems 6.1 and 12.1]. This result could also have been distilled from the classification of Banieqbal [Ban88], who classified all finite subgroups of 2 × 2-matrices over division rings of characteristic zero. We list the groups in the next theorem using their GAP identification number. Theorem 1.10.3 (Nebe) The maximal finite subgroups of 2×2-matrices over totally definite quaternion algebras D with center Q, spanning M2(D) over Q, are: [144, 124], [144, 128], [240, 89], [240, 90], [288, 389], [720, 409], [1152, 155468] and [1920, 241003].

Alternatively, Eric Jespers and Angel´ del R´ıowere able to obtain this list of finite subgroups of 2×2-matrices over totally definite quaternion algebras with center Q in [JdR], by describing the Sylow subgroups of such groups. This can be done only by using strong Shoda pairs and some elementary techniques. We conclude by studying finite subgroups of U(B) for exceptional components B of group algebras. Let G be a finite subgroup of U(B). When B is a division ring, G appears in Amitsur’s classification,√ see Theorem 1.10.1. When B is an exceptional 2 × 2-matrix ring M2(Q( −d)) with d ∈ {0, 1, 2, 3}, Ä −1,−1 ä G is a subgroup of GL(2, 25) by Proposition 1.9.7. If B is one of M2 , Q Ä −1,−3 ä Ä −2,−5 ä M2 or M2 , then G appears as a subgroup of the groups in Q Q the classification of Gabriele Nebe, see Theorem 1.10.3.

1.11 cyclotomic units

We are interested in U(ZG), and because of the Wedderburn decomposition, also in U(Z[ζn]). In this section, we deal with units in Z[ζn]. If n > 1 and k is an integer coprime to n, then one deﬁnes

k 1 − ζn 2 k−1 ηk(ζn) = = 1 + ζn + ζn + ··· + ζn , 1 − ζn

and one can check that ηk(ζn) is a unit of Z[ζn]. One extends this notation by deﬁning

ηk(1) = 1.

33 preliminaries

j The units of the form ηk(ζn), with j, k and n integers, such that gcd(k, n) = 1, are called the cyclotomic units of Q(ζn). The next lemma is easy to verify. Lemma 1.11.1

Let n > 1 and both k and k1 coprime with n. The cyclotomic units satisfy the following equalities:

(1) ηk(ζn) = ηk1 (ζn), if k ≡ k1 mod n;

k (2) ηkk1 (ζn) = ηk(ζn)ηk1 (ζn);

(3) η1(ζn) = 1;

−k (4) ηn−k(ζn) = −ζn ηk(ζn).

A classical result, which goes back to the work of Ernst Eduard Kummer from the 19th century, states that the cyclotomic units of Q(ζn) generate a subgroup of ﬁnite index in U(Z[ζn]), see for example [Was82, p. 147]. Further- more, when n is a power of a prime number, one knows the following basis (see [Was82, Theorem 8.2]). Theorem 1.11.2 (Kummer) Let p be a prime number and n a non-negative integer. The set

ß pn ™ η (ζ n ) : 1 < k < , p k k p 2 -

generates a free abelian subgroup of ﬁnite index in U(Z[ζpn ]).

The cyclotomic units of QG are, by deﬁnition, all the elements of QG which project to a cyclotomic unit in a commutative Wedderburn component Q(ζn) for some n ∈ N and project to 1 in the remaining components.

1.12 bass units

Consider a ﬁnite group G, an element g ∈ G of order n, and suppose that k and m are positive integers for which km ≡ 1 mod n. One veriﬁes that

1 − km u (g) = (1 + g + ··· + gk−1)m + (1 + g + g2 + ··· + gn−1) k,m n

34 1.12 bass units

is a unit of the integral group ring ZG. The units of this form were introduced by Hyman Bass in [Bas66] and are known as Bass units or Bass cyclic units. Note that Bass units of Z hgi project to powers of cyclotomic units in the Wedderburn components of Q hgi. We denote by B1(G) the group generated by the Bass units of ZG. Bass proved that, if G is a cyclic group, then B1(G) is a subgroup of finite index in U(ZG). Hyman Bass and John Willard Milnor extended this result to finite abelian groups using K-theory to reduce to the S cyclic case. If R is a ring, then GL(R) is defined as n≥1 GLn(R) and E(R) as S n≥1 En(R). By K1(R) one denotes GL(R)/E(R). A deep result of Hyman Bass proves that the rank of K1(ZG) equals the rank of Z(U(ZG)) and that the image of B1(G) in K1(ZG) is of finite index in K1(ZG) [Bas66, Corollary 6.3, Theorem 5]. Applied to finite abelian groups, this gives the following. Theorem 1.12.1 (Bass-Milnor)

When G is a finite abelian group, the group B1(G) is a subgroup of finite index in U(ZG). We will refer to this result as the Bass-Milnor Theorem while the Bass Theorem refers to the result for cyclic groups. The Bass Theorem also provides a basis consisting of Bass units for a free abelian subgroup of finite index in U(ZG), provided that G is cyclic. As far as we know, no basis consisting of Bass units for a free abelian subgroup of finite index was known for an arbitrary abelian group G. In Chapter 4, we give a constructive K-theoretic-free proof of the Bass-Milnor Theorem and describe a basis formed by Bass units for a free abelian subgroup of finite index in U(ZG) for finite abelian groups G. Jairo Gon¸calves and Donald Passman showed the following equalities in [GP06, Lemma 3.1], inspired by those for cyclotomic units from Lemma 1.11.1.

Lemma 1.12.2 (Gon¸calves-Passman) m Let g ∈ G, n = |g| and k, k1, m, m1 be positive integers such that k ≡ m m1 k1 ≡ k ≡ 1 mod n. The Bass units satisfy the following equalities:

(1) uk,m(g) = uk1,m(g), if k ≡ k1 mod n;

(2) uk,m(g)uk,m1 (g) = uk,m+m1 (g);

k (3) uk,m(g)uk1,m(g ) = ukk1,m(g);

(4) u1,m(g) = 1;

35 preliminaries

−m (5) u−1,m(g) = (−g) ;

i (6) uk,m(g) = uk,mi(g), for a non-negative integer i;

−1 k (7) uk,m(g) = uk1,m(g ), if kk1 ≡ 1 mod n;

−km m (8) un−k,m(g) = uk,m(g)g provided (−1) ≡ 1 mod n.

By (1), one can allow negative integers k with the obvious meaning and by (6) and (7), an integral power of a Bass unit is a Bass unit. Let N be a normal subgroup of G. Using equations (1) and (6) together with the Chinese Remainder Theorem, it is easy to verify that some power of a Bass unit in Z(G/N) is the natural image of a Bass unit in ZG. By (4) and (5), it is clear that uk,m(g) is of finite order if k ≡ ±1 mod |g|. The opposite is proven in [PMS02, Proposition 8.1.12]. Proposition 1.12.3 Consider a finite group G, an element g ∈ G of order n and suppose that k and m are positive integers for which km ≡ 1 mod n. The Bass unit uk,m(g) has finite order if and only if k ≡ ±1 mod |g|.

1.13 bicyclic units

There are not many recipes known for constructing units in group rings. Be- sides the famous Bass units, the construction of bicyclic units, introduced by J¨urgenRitter and Sudarshan Sehgal in [RS91b], is also well known. The bicyclic units of ZG are the elements of one of the following forms

2 n−1 βg,h = 1 + (1 − g)h(1 + g + g + ··· + g )

and 2 n−1 γg,h = 1 + (1 + g + g + ··· + g )h(1 − g), where g, h ∈ G and g has order n. We denote by B2(G) the group generated by the bicyclic units of ZG. More generally, given a collection {e1, . . . , es} of idempotents of QG, one deﬁnes generalized bicyclic units

β = 1 + z2 (1 − e )he ei,h ei i i

36 1.13 bicyclic units

and γ = 1 + z2 e h(1 − e ), ei,h ei i i with h ∈ G and where zei ∈ N is chosen of minimal value with respect to the

property that zei ei lies in ZG. One calls the group generated by the various

βei,h and γei,h the group of generalized bicyclic units and denotes it by GB2(G). Note that, when using GB2(G), a collection of idempotents should be given at least implicitly. The idea of the following lemma comes from [JL93]. Lemma 1.13.1 (Jespers-Leal)

Let G be a ﬁnite group and {e1, . . . , em} a collection of primitive central idempotents of QG. Consider the set {geb i : g ∈ G, i ∈ {1, . . . , m}} of idempotents of QG. Then GB2(G) ⊆ B2(G).

Proof We will prove that γgeˆ i,h ∈ B2(G). Analogously, βgeˆ i,h ∈ B2(G) and the result follows. Take g, h ∈ G with |g| = n, and for i ∈ {1, . . . , m}, take zi minimal such that z nge ∈ G. Consider γ = 1+z2n2ge h(1−ge ). Let α = Pn−1(n−i)gi ∈ i b i Z geˆ i,h i b i b i i=0 ZG. It is clear that gb(1 − g) = 0. Easy computations verify that α(1 − g) = n(1 − g) and γ = 1 + z2nghe α(1 − g). b geˆ i,h i b i Because of the equation

k l (1 + ngxb (1 − g)) (1 + ngyb (1 − g)) = 1 + ngb(kx + ly)(1 − g), it easily follows that

{1 + ngαb (1 − g): α ∈ ZG} ⊆ B2(G).

Hence, γgeˆ i,h ∈ B2(G) and the proof is ﬁnished. The following proposition is a reformulation of a result of Chapter 22 from Sudarshan Sehgal’s book [Seh93]. Proposition 1.13.2 Assume G is a ﬁnite group and let U ⊆ U(ZG) be a subgroup of the unit Ln Ln group of the integral group ring. Let QG = i=1 QGei = i=1 Mni (Di)

be the Wedderburn decomposition of QG and let ZGei = Mni (Oi) denote the orders in each simple component.

37 preliminaries

Then U is of ﬁnite index in U(ZG) if and only if both of the following hold:

1. The natural image of U in K1(ZG) is of ﬁnite index; 2. For each i ∈ {1, . . . , n}, the group U contains a subgroup of ﬁnite

index in 1 − ei + SLni (Oi).

By Theorem 1.12.1, one already knows that the natural image of B1(G) in K1(ZG) is of finite index. This means that, in order to find generators for a subgroup of finite index, one needs to look for units to enlarge B1(G) to a set U such that the second condition is satisfied as well. If QG does not contain exceptional components, then one knows that in each Wedderburn component, the elementary matrices En(I), for any non-zero ideal I of O, generate a subgroup of finite index in SLn(O), see Theorems 1.9.2 and 1.9.3. In this case, the usual method is to find units which behave similar as the elementary matrices in each Wedderburn component of QG. For several classes of finite groups G including nilpotent groups of odd order, JürgenRitter and Sudarshan Sehgal [RS89, RS91b, RS91a] showed that B2(G) fulfills the second condition and hence B1(G) ∪ B2(G) generates a subgroup of finite index in U(ZG). Let e be a primitive central idempotent of QG such that QGe = Mn(D) is not exceptional and let O be a maximal order in D. Eric Jespers and Guil- herme Leal proved in [JL93, Proposition 3.2] that if f is a non-central idempotent in QGe, then the group generated by the generalized bicyclic units based on f, does contain En(I) for some non-zero ideal I of O. This means that B1(G) ∪ GB2(G) will always satisfy the conditions of Proposition 1.13.2 whenever QG does not contain exceptional components and if one can construct a non-central idempotent in each non-commutative Wedderburn component of QG. The following result is a reformulation of [JL93, Theorem 3.3]. Proposition 1.13.3 (Jespers-Leal) Ln Let G be a finite group such that QG = i=1 QGei does not contain exceptional components. Let f1, . . . , fk be a collection of idempotents in QG such that for each i ∈ {1, . . . , n} contributing to a non-commutative component QGei, there is some index j(i) such that eifj(i) is non-central in QGei. The set B1(G) ∪ GB2(G) generates a subgroup of finite index in U(ZG).

38 1.14 central units

If G has no non-abelian homomorphic image which is ﬁxed point free, then all projections Gei into the non-commutative components QGei, yield a non- central idempotent g“iei, for some gi ∈ G. The next result follows immediately from Lemma 1.13.1 and Proposition 1.13.3, see [JL93, Corollary 4.1].

Corollary 1.13.4 (Jespers-Leal) Let G be a finite group such that QG does not contain exceptional components. If G has no non-abelian homomorphic image which is fixed point free, then the set B1(G) ∪ B2(G) generates a subgroup of finite index in U(ZG).

1.14 central units

Consider a ﬁnite group G and two of its elements g, h ∈ G. Recall that g and h are said to be R-conjugate (respectively, Q-conjugate) if g is conjugate to either h or h−1 (respectively, to hr for some integer r coprime with the order of h) within G. This deﬁnes two equivalence relations on G and their equivalence classes are called R-classes and Q-classes of G, respectively. The following theorem is due to Samuil D. Berman and Ernst Witt and can be found in [CR62, Theorem 42.8].

Theorem 1.14.1 (Berman-Witt) The number of R- (respectively, Q-) classes of G coincides with the number of simple components of the Wedderburn decomposition of RG (respectively, QG).

Let Z(U(ZG)) denote the group of central units in the integral group ring ZG, for G a finite group. By Theorem 1.8.7, one knows that Z(U(ZG)) is finitely generated. Moreover, inspired by Graham Higman, one deduces that Z(U(ZG)) is equal to ±Z(G) × T , where T is a finitely generated free abelian subgroup of Z(U(ZG)), see [PMS02, Corollary 7.3.3]. Using the theorem of Berman and Witt and Dirichlet’s Unit Theorem 1.8.8, one can prove that the rank of Z(U(ZG)) for G a finite group is the difference between the number of R-classes and Q-classes of G. From this, one deduces the rank of Z(U(ZG)) in terms of the number of Wedderburn components, as done, independently, by JürgenRitter and Sudarshan Sehgal [RS05, Theorem 2], and Raul Antonio Ferraz [Fer04, Theorem 3.6].

39 preliminaries

Theorem 1.14.2 (Ritter-Sehgal-Ferraz ) Let G be a finite group. Then Z(U(ZG)) is finitely generated and its rank equals the difference between the number of simple components of RG and the number of simple components of QG. When G is an abelian group, the formula above simplifies as shown by Graham Higman [Hig40]. The exact formula was given by Raymond G. Ayoub and Christine Ayoub in [AA69]. Theorem 1.14.3 (Higman) Let G be a finite abelian group. Then U(ZG) is finitely generated abelian 1+k2+|G|−2c and has rank r = 2 , where c is the number of cyclic subgroups of G and k2 is the number of elements of G of order 2.

40 2

WEDDERBURNDECOMPOSITIONANDIDEMPOTENTS

A concrete realization of the Wedderburn decomposition of semisimple group algebras by means of its central idempotents and orthogonal primitive idempotents is of importance for many topics. For example, Jespers and Leal have shown one of the profits of idempotents to units of ZG in Proposition 1.13.3. In this chapter, we extend the results from Theorem 1.7.9 and Proposi- tion 1.7.13, due to Aurora Olivieri, Angel´ del R´ıoand Juan Jacobo Simón,to group algebras FG over number fields F . Secondly, we describe a complete set of orthogonal primitive idempotents of QG and of finite semisimple group algebras FG, when G is a finite strongly monomial group such that each strong Shoda pair yields a trivial twisting τ.

2.1 the wedderburn decomposition of FG

In this section, we assume G to be a ﬁnite group and F to be a number ﬁeld. All character of G are assumed to be complex characters.

If G = hgi is cyclic of order k, then the irreducible characters are all linear and are deﬁned by the image of a generator of G. Therefore the set G∗ = Irr(G) of irreducible characters of G is a group and the map ψ : Z/kZ → G∗ m ∗ given by ψ(m)(g) = ζk is a group homomorphism. The generators of G are precisely the faithful representations of G. Let CF (G) denote the set of orbits of the faithful characters of G under the action of Gal(F (ζk )/F ). Note that for any faithful character χ of G, F (χ) = F (ζk ).

41 wedderburn decomposition and idempotents

Each automorphism σ ∈ Gal(F (ζk )/F ) is completely determined by its t image of ζk , and is given by σ(ζk ) = ζk , where t is an integer uniquely determined modulo k. In this way, one gets the following morphisms Gal(F (ζk )/F ) / Gal(Q(ζk )/Q)

' ' Ik (F ) / U (Z/kZ) where we denote the image of Gal(F (ζk )/F ) in U (Z/kZ) by Ik (F ). In this setting, the sets in CF (G) correspond one-to-one to the orbits under the action of Ik (F ) on U (Z/kZ) by multiplication. Let N ¢ G be such that G/N is cyclic of order k and C ∈ CF (G/N ).

If χ ∈ C and tr = trF (ζk )/F denotes the ﬁeld trace of the Galois extension F (ζ )/F (i.e. tr(x) = P σ(x) for all x ∈ F (ζ )), then we k σ∈Gal(F (ζk )/F ) k set 1 X ε (G,N ) = tr(χ(gN ))g −1 C |G| g∈G 1 X X = ψ(gN )g −1 |G| g∈G ψ∈C −1 X X −1 = [G : N ] N“ ψ(X )gX , X ∈G/N ψ∈C

where gX denotes a representative of X ∈ G/N . Note that εC (G,N ) does not depend on the choice of χ ∈ C. Let K ¢ H ≤ G such that H/K is cyclic and C ∈ CF (H/K ). Then eC (G,H,K ) denotes the sum of the different G-conjugates of εC (H,K ). Proposition 2.1.1 If G is a finite abelian group of order n and F is a number field, then the map (N,C) 7→ εC (G, N) is a bijection from the set of pairs (N,C) with N ¢ G, such that G/N is cyclic, and C ∈ CF (G/N) to the set of primitive central idempotents of FG. Furthermore, for every N ¢ G and C ∈ CF (G/N), F GεC (G, N) ' F (ζk), where k = [G : N].

Proof If e is a primitive central idempotent of FG, then there exists an irreducible character ψ of G such that e = eF (ψ). Since G is abelian, ψ is

42 2.1 the wedderburn decomposition of FG

linear. Let N denote the kernel of ψ and let χ be the faithful character of G/N given by χ(gN) = ψ(g). Then G/N is cyclic, say of order k, F (ψ) = F (ζk) and the orbit of χ under the action of Gal(F (ζk)/F ) is an element of CF (G/N). Furthermore X eF (ψ) = σ · e(ψ) σ∈Gal(F (ψ)/F ) 1 X X = σ(ψ(g))g−1 |G| σ∈Gal(F (ψ)/F ) g∈G 1 X X = σ(χ(gN))g−1 (4) |G| σ∈Gal(F (ψ)/F ) g∈G 1 X = tr (χ(gN))g−1 |G| F (ζk)/F g∈G

= εC (G, N).

This shows that the map is surjective and, since neF (ψ) = eF (ψ) for all n ∈ N, F GεC (G, N) = F GeF (ψ) ' F (ζk).

Assume now that εC1 (G, N1) = εC2 (G, N2) with Ni ¢G and Ci ∈ CF (G/Ni). Take χi ∈ Ci. Let πi : G → G/Ni be the canonical projections and ψi = χi ◦πi.

Then ψi are irreducible characters of G. By (4), eF (ψ1) = εC1 (G, N1) =

εC2 (G, N2) = eF (ψ2) and F (ψ1) = F (ψ2). If K = F (ψi), then there exists a σ ∈ Gal(K/F ) such that ψ2 = σ ◦ ψ1 and hence N1 = ker ψ1 = ker ψ2 = N2. −1 −1 −1 Now let π be a right inverse of π1 = π2. Then χ2 = ψ2 ◦π = σ◦ψ1 ◦π = σ ◦ χ1 and hence C1 = C2. This shows that the map is injective. Corollary 2.1.2 If G is a ﬁnite group with normal subgroup N such that G/N is cyclic of order k and F is a number ﬁeld, then εC (G, N) is a primitive central idempotent of FG and F GεC (G, N) ' F (ζk). Furthermore, if D ∈ CF (G/N), then εC (G, N) = εD(G, N) if and only if C = D.

Proof The natural epimorphism G → G/N induces a ring isomorphism ψ : FGN“ ' F (G/N) (see (1) Section 1.6). Since εC (G, N) ∈ FGN“ and ψ(εC (G, N)) = εC (G/N, 1) is a primitive central idempotent of F (G/N) by Proposition 2.1.1, also εC (G, N) is a primitive central idempotent in FG. Moreover, again by Proposition 2.1.1, F GεC (G, N) ' F (G/N)εC (G/N, 1) ' F (ζk) and εC (G, N) = εD(G, N) if and only if εC (G/N, 1) = εD(G/N, 1) if and only if C = D.

43 wedderburn decomposition and idempotents

Let H be a subgroup of G, ψ a linear character of H and g ∈ G. Then ψg denotes the character of Hg given by ψg(hg) = ψ(h). Since ker(ψg) = ker(ψ)g, the map ψ 7→ ψg is a bijection between linear characters of H with kernel K and linear characters of Hg with kernel Kg. This map induces a bijection g g g CF (H/K) → CF (H /K ): C 7→ C . In this sense, the following equation is easy to verify:

g g g εC (H,K) = εCg (H ,K ). (5)

Now let K ¢ H ≤ G be such that H/K is cyclic of order k. Then N = NG(H) ∩ NG(K) acts on CF (H/K) by the rule

i g i {χ : i ∈ Ik(F )}· g = {(χ ) : i ∈ Ik(F )}.

Consider the stabilizer of a C ∈ CF (H/K), take χ ∈ C and ﬁx a generator hK of H/K,

g i StabN (C) = {g ∈ N : χ = χ for some i ∈ Ik(F )} −1 i = {g ∈ N : χ(g hgK) = χ(h K) for some i ∈ Ik(F )} −1 i = {g ∈ N : g hgK = h K for some i ∈ Ik(F )}.

Note that we used that χ is faithful in the third equality. Note that StabN (C) is independent on the choice of generator of H/K. Indeed, let h1K and h2K j be generators such that h1K = h2K. Then

−1 i StabN (C) = {g ∈ N : g h1gK = h1K for some i ∈ Ik(F )} −1 j i j = {g ∈ N :(g h2gK) = (h2K) for some i ∈ Ik(F )} −1 i ⊇ {g ∈ N : g h2gK = h2K for some i ∈ Ik(F )}, which gives an equality when reversing the role of h1 and h2. Hence StabN (C) is independent on the choice of C ∈ CF (H/K) and hence it is the stabilizer of any C ∈ CF (H/K) under the action of N. We denote this stabilizer by

EF (G, H/K). Note that Ik(Q) = U(Z/kZ) and EQ(G, H/K) = N. By Proposition 1.7.8, all irreducible characters of a monomial group G are associated with Shoda pairs of G. The next theorem provides a description of the primitive central idempotent given by a Shoda pair (H,K) of a group G as a multiple of eC (G, H, K).

44 2.1 the wedderburn decomposition of FG

Theorem 2.1.3 Let G be a ﬁnite group, (H,K) a Shoda pair of G and F be a number ﬁeld. Let χ be a linear character of H with kernel K and C be the orbit of χ under the action of Gal(F (χ)/F ). Then χG is irreducible and the primitive central idempotent of FG associated to χG is

[Cen (ε (H,K)) : H] e (χG) = G C e (G, H, K). F [F (χ): F (χG)] C

Proof Let e = e(χ). Let Gal(F (χ)/F ) = {σ1, . . . , σn} and T = {g1, . . . , gm} be a right transversal of H in G. Then

1 X e(χG) = χG(1)χG(g−1)g |G| g∈G m ! 1 X |G| X = χ(1) χ◦(g g−1g−1) g |G| |H| i i g∈G i=1 m 1 X X = χ(h−1)g−1hg |H| i i i=1 h∈H m X = e · gi. i=1

Consider the table

G σ1 · e · g1 σ1 · e · g2 ··· σ1 · e · gm σ1 · e(χ ) G σ2 · e · g1 σ2 · e · g2 ··· σ2 · e · gm σ2 · e(χ ) ············ ··· G σn · e · g1 σn · e · g2 ··· σn · e · gm σn · e(χ ) εC (H,K) · g1 εC (H,K) · g2 ··· εC (H,K) · gm ∗

We can compute the total sum ∗ by adding the elements of the last column or the elements of the last row:

n m X G X ∗ = σi · e(χ ) = εC (H,K) · gj. (6) i=1 j=1

45 wedderburn decomposition and idempotents

In the ﬁrst sum of (6) the elements to add are the elements of the orbit of e(χG) under the action of Gal(F (χ)/F ), each of them repeated [F (χ): F (χG)] times. Using the formula of Yamada (2) from Section 1.7, one has

G G ∗ = [F (χ): F (χ )]eF (χ ). (7)

Similarly the second sum of (6) adds up the elements of the G-orbit of εC (H,K) by (5), each of them repeated [CenG(εC (H,K)) : H] times. Therefore

∗ = [CenG(εC (H,K)) : H]eC (G, H, K). (8)

The theorem follows by comparing (7) and (8).

So for each Shoda pair (H,K) of G, number ﬁeld F and each C ∈ CF (H/K), there exists a unique α ∈ Q such that αeC (G, H, K) is a primitive central idempotent of FG. Next, we will investigate the case when α = 1. This happens when (H,K) is a strong Shoda pair of G.

Lemma 2.1.4 Let F be a number ﬁeld. Let K ¢ H ≤ G be such that H/K is cyclic of order k and C ∈ CF (H/K). 1. For every g ∈ G, the following statements are equivalent: a) g ∈ K;

b) gεC (H,K) = εC (H,K); c) gεb C (H,K) = εC (H,K).

2. If H ¢ NG(K), then CenG(εC (H,K)) = EF (G, H/K).

Proof 1. The fact that 1a) implies 1b) follows from the easy observation that gK“ = K“ when g ∈ K. The equivalence between 1b) and 1c) follows by comparing the coeﬃcients. It remains to prove that g ∈ K when gεC (H,K) = εC (H,K). Assume that gεC (H,K) = εC (H,K). Because of Corollary 2.1.2, εC (H,K) is a primitive central idempotent in FH and hence

46 2.1 the wedderburn decomposition of FG

non-zero. Therefore, the support of gεC (H,K) is a non-empty set in H and g ∈ H. Hence we can write g = xht for some x ∈ K and 0 ≤ t ≤ k. Now

k−1 −1 X X i t−i k K“ ψ(h K)h = gεC (H,K) i=0 ψ∈C

= εC (H,K) k−1 X X = k−1K“ ψ(hiK)h−i i=0 ψ∈C and hence

t i X t i trF (ζk)/F ((χ(h K) − 1)χ(h K)) = (ψ(h K) − 1)ψ(h K) = 0 ψ∈C for every 0 ≤ i ≤ k−1 and some χ ∈ C. Since χ is faithful, its image generates

F (χ) = F (ζk) as F -vector space. Since trF (ζk)/F : F (ζk) → F is F -linear and surjective, we deduce that χ(htK) = 1 and hence k divides t. Therefore t = 0 and g ∈ K. 2. When H ¢ NG(K), clearly EF (G, H/K) ⊆ NG(H) ∩ NG(K) ⊆ NG(K). Furthermore, if g ∈ CenG(εC (H,K)), then for each x ∈ K

−1 −1 g xgεC (H,K) = g xεC (H,K)g = εC (H,K),

−1 by 1. Therefore g xg ∈ K and CenG(εC (H,K)) ⊆ NG(K). Take g ∈ NG(K). By Corollary 2.1.2, εC (H,K) and εCg (H,K) are two primitive central idempo- g tents of FH and they are equal if and only if C = C (i.e. if g ∈ EF (G, H/K)). g By (5), εC (H,K) = εCg (H,K). Hence g ∈ CenG(εC (H,K)) if and only if g εC (H,K) = εC (H,K), if and only if g ∈ EF (G, H/K). Clearly, elements of QG can also be seen as elements in FG. The following lemma tells how ε(G, N) and e(G, H, K) can be written as a sum of elements in FG. Lemma 2.1.5 Let F be a number ﬁeld.

1. Let N ¢ G be such that G/N is cyclic, then

X ε(G, N) = εC (G, N).

C∈CF (G/N)

47 wedderburn decomposition and idempotents

2. Let K ¢ H ¢ NG(K) be such that H/K is cyclic and let R be a set of representatives of the action of NG(K) on CF (H/K). Then X e(G, H, K) = eC (G, H, K). C∈R

Proof 1. For every C ∈ CF (H/K), both ε(G, N) and εC (G, N) belong to FGN“. By factoring out N and using the isomorphism FGN“ ' F (G/N), we may assume without loss of generality that N = 1 and G is cyclic. By Proposition 2.1.1, every primitive central idempotent of FG is of the form εC (G, H) with H ¢G and C ∈ CF (G/H). Therefore ε(G, 1) is the sum of some elements εC (G, H) and it is enough to prove that if H ¢ G and C ∈ CF (G/H), then ε(G, 1)εC (G, H) 6= 0 if and only if H = 1. If C ∈ CF (G) and 1 6= x ∈ G, then (1 − xb)εC (G, 1) 6= 0 by Lemma 2.1.4. Since εC (G, 1) is a primitive central idempotent, we have that εC (G, 1) = (1 − xb)εC (G, 1). Since ε(G, 1) is the product of elements of the form 1 − xb with 1 6= x ∈ G, it follows that ε(G, 1)εC (G, 1) = εC (G, 1) 6= 0. Conversely, if 1 6= H ≤ G, then there exists a h ∈ H such that hhi is a minimal non-trivial subgroup of G. Hence ε(G, 1)εC (G, H) = ε(G, 1)(1 − bh)εC (G, H) = 0, by Lemma 2.1.4. This ﬁnishes part 1 of the proof. 2. Let N = NG(H) ∩ NG(K) = NG(K), E = EF (G, H/K), TN be a right transversal of N in G, TE be a right transversal of E in N. Then {hg : h ∈ TE, g ∈ TN } is a right transversal of E in G. By Proposition 1.7.12, we know that N = Cen (ε(H,K)). Hence e(G, H, K) = P ε(H,K)g. G g∈TN t Clearly, CF (H/K) is the disjoint union of the sets {C : t ∈ TE} for C running on R. Therefore, X e(G, H, K) = ε(H,K)g

g∈TN X X g = εC (H,K)

g∈TN C∈CF (H/K) X X X g = εCh (H,K)

g∈TN C∈R h∈TE X X X hg = εC (H,K)

C∈R g∈TN h∈TE X = eC (G, H, K). C∈R

48 2.1 the wedderburn decomposition of FG

In the main theorem of this section we describe the simple components of the group algebra FG provided by strong Shoda pairs. We show that a strong Shoda pair (H,K) of G that determines a primitive central idempotent e(G, H, K) in QG, will also determine a primitive central idempotent eC (G, H, K) in FG for C ∈ CF (G/N).

Theorem 2.1.6 Let G be a ﬁnite group and F be a number ﬁeld.

1. Let (H,K) be a strong Shoda pair of G and C ∈ CF (H/K). Let [H : K] = k, yK a generator of H/K and E = EF (G, H/K). Then eC (G, H, K) is a primitive central idempotent of FG and

σ F GeC (G, H, K) ' M[G:E] (F (ζk) ∗τ E/H) ,

where σ and τ are deﬁned as follows. Let ψ : E/H → E/K be a left inverse of the projection E/K → E/H. Then

i ψ(gH) i σgH (ζk) = ζk, if yK = y K, 0 j 0 −1 0 j τ(gH, g H) = ζk, if ψ(gg H) ψ(gH)ψ(g H) = y K,

for gH, g0H ∈ E/H and integers i and j.

2. Let X be a set of strong Shoda pairs of G. If every primitive central idempotent of QG is of the form e(G, H, K) for (H,K) ∈ X, then every primitive central idempotent of FG is of the form eC (G, H, K) for (H,K) ∈ X and C ∈ CF (H/K).

Proof 1. By Lemma 2.1.4, E = CenG(εC (H,K)) and by Proposition 1.7.12, CenG(ε(H,K)) = NG(K). Let T be a right transversal of E in G, then P g eC (G, H, K) = g∈T εC (H,K) .

In order to prove that eC (G, H, K) is an idempotent, it is enough to show that the G-conjugates of εC (H,K) are orthogonal. For this we show that, if g g ∈ G \ E, then εC (H,K)εC (H,K) = 0. By Lemma 2.1.5,

g g g εC (H,K)εC (H,K) = εC (H,K)ε(H,K)ε(H,K) εC (H,K) ,

49 wedderburn decomposition and idempotents

which is zero by the deﬁnition of a strong Shoda pair whenever g∈ / NG(K). g If g ∈ NG(K) \ E, then εC (H,K) = εCg (H,K) 6= εC (H,K) are two diﬀerent primitive central idempotents of FH by Corollary 2.1.2. Hence

g εC (H,K)εC (H,K) = 0.

By Corollary 2.1.2, F HεC (H,K) ' F (ζk). Also, by Lemma 2.1.4,

σ F EεC (H,K) = F HεC (H,K) ∗τ E/H

is a crossed product with homogeneous basis ψ(E/H), where ψ : E/H → E/K is a left inverse of the projection E/K → E/H. The action σ and twisting τ are given by

σ : E/H → Aut(F HεC (H,K)) −1 gH 7→ (αεC (H,K) 7→ ψ(gH) αεC (H,K)ψ(gH)),

τ : E/H × E/H → U(F HεC (H,K)) (gH, g0H) 7→ ψ(gg0H)−1ψ(gH)ψ(g0H).

Clearly, the isomorphism F HεC (H,K) ' F (ζk) extends to an E/H-graded isomorphism

σ σ0 F EεC (H,K) = F HεC (H,K) ∗τ E/H ' F (ζk) ∗τ 0 E/H.

Since H/K is maximal abelian in N/K and hence also in E/K, the action σ0 is faithful and F EεC (H,K) is a simple algebra (Theorem 1.5.1). If g ∈ G, then the map x 7→ xg deﬁnes an isomorphism between the FG- g modules F GεC (H,K) and F GεC (H,K) . Therefore,

M t [G:E] F GeC (G, H, K) = F GεC (H,K) ' (F GεC (H,K)) , t∈T

as FG-modules. Moreover, M εC (H,K)F GεC (H,K) = εC (H,K)F EtεC (H,K) t∈T M = F EεC (H,K)tεC (H,K) t∈T

= F EεC (H,K),

50 2.1 the wedderburn decomposition of FG

t because εC (H,K) is central in FE and εC (H,K) εC (H,K) = 0 for all t ∈ G \ E. Thus

F GeC (G, H, K) ' EndFG(F GeC (G, H, K))

' M[G:E](EndFG(F GεC (H,K)))

' M[G:E](εC (H,K)F GεC (H,K))

' M[G:E](F EεC (H,K)) σ0 ' M[G:E](F (ζk) ∗τ 0 E/H).

2. By assumption there is a set Y ⊆ X such that {e(G, H, K):(H,K) ∈ Y } is the set of primitive central idempotents of QG. Hence, by Lemma 2.1.5,

X X X 1 = e(G, H, K) = eC (G, H, K),

(H,K)∈Y (H,K)∈Y C∈R(H,K)

for R(H,K) a set of representatives of the action of NG(K) on CF (H/K). Furthermore, eC (G, H, K) are primitive central idempotents of FG by part 1. Hence {eC (G, H, K):(H,K) ∈ Y,C ∈ R(H,K)} is a complete and non- redundant set of primitive central idempotents of FG.

Applying Proposition 1.7.14, we get the following result.

Corollary 2.1.7 If G is a ﬁnite strongly monomial group and F is a number ﬁeld, then every primitive central idempotent of FG is of the form eC (G, H, K) for a strong Shoda pair (H,K) of G and C ∈ CF (H/K). Furthermore, for every strong Shoda pair (H,K) of G and every C ∈ CF (H/K),

σ F GeC (G, H, K) ' M[G:E] F ζ[H:K] ∗τ E/H ,

where σ and τ are deﬁned as in Theorem 2.1.6 and E = EF (G, H/K).

Applying Theorem 1.7.16, we get the following result.

51 wedderburn decomposition and idempotents

Corollary 2.1.8 If G is a ﬁnite metabelian group, A a maximal abelian subgroup of G containing G0 and F a number ﬁeld. Then every primitive central idempotent of FG is of the form eC (G, H, K) for a pair (H,K) of subgroups of G satisfying the following conditions:

1. H is a maximal element in {B ≤ G : A ≤ B and B0 ≤ K ≤ B};

2. H/K is cyclic;

and C ∈ CF (H/K).

Example 2.1.9 Consider the nilpotent (and thus strongly monomial) group G = C3 × Q8 with presentation

a, x, y : a3 = 1, x4 = 1, y2 = x2, y−1xy = x−1, ax = xa, ay = ya .

Using the GAP-package wedderga [BHK+14], one can compute the following Wedderburn decompositions: Å−1, −1ã QG = 4Q ⊕ 4Q(ζ3) ⊕ ⊕ M2(Q(ζ3)), Q Q(ζ3)G = 12Q(ζ3) ⊕ 3M2(Q(ζ3)). The corresponding strong Shoda pairs are

(G, G), (G, ha, xi), (G, ha, yi), (G, a, xy−1 ), (G, hx, yi), (G, hxi), (G, hyi), (G, xy−1 ), (ha, xi , hai) and (ha, xi , 1). (9)

The last two pairs contribute to the non-commutative components of QG and Q(ζ3)G. Let H = ha, xi and K = 1. The strong Shoda pair (H,K) realizes the component M2(Q(ζ3)) of QG and hence also the components

Q(ζ3) ⊗Q M2(Q(ζ3)) = 2M2(Q(ζ3))

of Q(ζ3)G. We will show how to construct the corresponding idempotents. Let H and K be as before, then [H : K] = 12 and H/K ' C12. The unit group of Z/12Z equals {1, 5, 7, 11} and hence I12(Q) = {1, 5, 7, 11}. Con- sider now the Galois group Gal(Q(ζ12)/Q(ζ3)). This group contains precisely i the automorphisms determined by ζ12 7→ ζ12 with i ∈ {1, 7}. Therefore,

52 2.1 the wedderburn decomposition of FG

i I12(Q(ζ3)) = {1, 7}. From now on, we denote the map ζ12 7→ ζ12 by the integer i. We compute

C (H/K) = {{1, 5, 7, 11}} and C (H/K) = {{1, 7}, {5, 11}}. Q Q(ζ3)

Since CQ(H/K) contains only one element, the pair (H,K) leads to one primitive central idempotent of QG. Take C = {1, 5, 7, 11} ∈ CQ(H/K). Then, 1 X X ε (H,K) = ψ(h)h−1 C 12 h∈H ψ∈C 12 1 X X = ψ(xa)i(xa)−i 12 i=1 ψ∈C 12 1 X = (ζi + ζ5i + ζ7i + ζ11i)(xa)−i. 12 12 12 12 12 i=1

Since C (H/K) contains two elements, the pair (H,K) leads to two prim- Q(ζ3) itive central idempotents of Q(ζ3)G. Take C1 = {1, 7} and C2 = {5, 11} ∈ C (H/K), then Q(ζ3)

12 1 X ε (H,K) = (ζi + ζ7i )(xa)−i, C1 12 12 12 i=1 12 1 X ε (H,K) = (ζ5i + ζ11i)(xa)−i, and C2 12 12 12 i=1

εC (H,K) = εC1 (H,K) + εC2 (H,K).

−1 −1 7 Since y axy = ax = (ax) and 7 ∈ I12(Q(ζ3)), both EQ(G, H/K) and E (G, H/K) equal G. Therefore e (G, H, K) = ε (H,K), e (G, H, K) = Q(ζ3) C C C1

εC1 (H,K) and eC2 (G, H, K) = εC2 (H,K).

Consider the simple components QGeC (G, H, K), Q(ζ3)eC1 (G, H, K) and

Q(ζ3)eC2 (G, H, K). By Theorem 2.1.6, they are all isomorphic to the crossed P2 2 6 2 product Q(ζ12) ∗ hyHi = i=1 Q(ζ12)uyi , where uy = ζ12 = −1 since y = 2 6 −1 7 −1 −1 7 x = (ax) and uy ζ12uy = ζ12 since y axy = ax = (ax) . We can rewrite

2 2 −1 7 Q(ζ12) ∗ hyHi ' Q(ζ3)(i, u : i = −1, u = −1, u iu = i = −i) Å−1, −1ã ' . Q(ζ3)

53 wedderburn decomposition and idempotents

2 2 2 2 Since −ζ3 − (ζ3 ) = 1 is satisfied in Q(ζ3), this quaternion algebra splits by Proposition 1.2.1 and equals M2(Q(ζ3)). Remark 2.1.10 There is a strong correspondence between the simple components in semisimple finite group algebras and simple components in group n algebras over number fields. Let Fq be the finite field of order q = p then Fq(ζk) = Fqo for an integer k coprime to p and o = ok(q). The Galois group Gal(Fq(ζk)/Fq) is cyclic of order ok(q) and can be seen as a subgroup of U(Z/kZ). For a cyclic group G, the elements of CF(G) are also referred to as the q-cyclotomic classes of G. Hereby, we find back the following result from [BdR07, Theorem 7]. Corollary 2.1.11 (Broche-del R´ıo) Let G be a finite group and F a finite field of order q such that FG is semisimple. Let (H,K) be a strong Shoda pair of G and C ∈ CF(H/K). Then eC (G, H, K) is a primitive central idempotent of FG and

FGeC (G, H, K) ' M[G:H](Fqo/[E:H] ),

where E = EF(G, H/K) and o is the multiplicative order of q modulo [H : K].

Remark 2.1.12 From [BdR07, Theorem 7], one also knows that there is a strong relation between the primitive central idempotents in a rational group algebra QG and the primitive central idempotents in a finite semisimple group algebra FG which are realized by the strong Shoda pairs of G. More precisely, if X is a set of strong Shoda pairs of G and every primitive central idempotent of QG is of the form e(G, H, K) for (H,K) ∈ X, then every primitive central idempotent of FG is of the form eC (G, H, K) for (H,K) ∈ X and C ∈ CF(H/K). Corollary 2.1.13 If G is a strongly monomial group and F is a finite field of order q such that FG is semisimple, then every primitive central idempotent of FG is of the form eC (G, H, K) for (H,K) a strong Shoda pair of G and C ∈ CF(H/K).

Example 2.1.14 Consider again the group G = C3 × Q8 from Example 2.1.9. All primitive central idempotents of F5G are realized by the strong Shoda pairs from (9). Consider H = ha, xi and K = 1, then [H : K] = 12 and

54 2.2 primitive idempotents of QG

F5(ζ12) = F5o12(5) = F25. The Galois group of the extension F25/F5 is gen- 5 erated by the Frobenius automorphism x 7→ x . Hence I12(F5) = {1, 5}. −1 7 Since y axy = (ax) and 7 ∈/ I12(F5), EF5 (G, H/K) = H. Furthermore, CF5 (H/K) = {{1, 5}, {7, 11}}. Let C1 = {1, 5} and C2 = {7, 11}, then

−1 eC1 (G, H, K) = εC1 (H,K) + y εC1 (H,K)y

= εC1 (H,K) + εC2 (H,K)

= eC2 (G, H, K).

Also F5GeC1 (G, H, K) ' M2(F25).

2.2 primitive idempotents of QG

As mentioned in Example 1.7.20, it is known that the construction of the orthogonal idempotents in Theorem 1.7.19 cannot be extended to, for example, arbitrary finite metacyclic groups. We present an alternative construction to describe a complete set of orthogonal primitive idempotents for a class of finite strongly monomial groups containing the finite metacyclic groups Cq m o1 Cpn . In this section we will focus on simple components of QG of a finite group G which are determined by a strong Shoda pair (H,K ) such that the twist- 0 0 ing τ (gH, g H ) = 1 for all g, g ∈ NG (K ) (with notation as in Proposi- tion 1.7.13). For such a component, we describe a complete set of orthogonal primitive idempotents (and a complete set of matrix units). This construction is based on the isomorphism of Theorem 1.5.2 on classical crossed products with trivial twisting. Such a description, together with the description of the primitive central idempotent e = e(G,H,K ) determining the simple component, yields a complete set of irreducible modules. NG (K )/H Before we do so, we need a basis of Q(ζ[H :K ] )/Q(ζ[H :K ] ) of the x form {w : x ∈ NG (K )/H } with w ∈ Q(ζ[H :K ] ). That such a basis exists follows from the Normal Basis Theorem 1.3.1. Theorem 2.2.1 Let (H,K) be a strong Shoda pair of a finite group G such that the 0 0 twisting τ(gH, g H) = 1 for all g, g ∈ NG(K). Let ε = ε(H,K) and e = e(G, H, K). Let F denote the fixed subfield of QHε under the natural action of NG(K)/H and [NG(K): H] = n. Let w be a normal element of QHε/F and B the normal basis determined by w. Let ψ be the isomorphism between QNG(K)ε and the matrix algebra Mn(F ) with

55 wedderburn decomposition and idempotents

respect to the basis B as stated in Theorem 1.5.2. Let P,A ∈ Mn(F ) be deﬁned as follows:  1 1 1 ··· 1 1   0 0 ··· 0 1   1 −1 0 ··· 0 0   1 0 ··· 0 0       1 0 −1 ··· 0 0   0 1 ··· 0 0  P =  ......  and A =  . . . . .  .  ......   . . . . .   ......   . . . . .   1 0 0 · · · −1 0   0 0 ··· 0 0  1 0 0 ··· 0 −1 0 0 ··· 1 0

Proof Consider the simple component

QGe ' M[G:N](QNε) ' M[G:N](QHε/F, 1) of QG with N = NG(K). Without loss of generality we may assume that K is normal in G and hence N = G. Indeed, if we obtain a complete set of orthogonal primitive idempotents of QNε, then the conjugates by the transversal T2 of N in G will give a complete set of orthogonal primitive idempotents of QGe since e = P εt and diﬀerent εt’s are orthogonal. t∈T2 gH From now on we assume that N = G and e = ε. Then B = {w : g ∈ T1}. Because (H,K) is a strong Shoda pair of G, the group H/K is a maximal abelian subgroup of NG(K)/K. Since G/H acts on QHe via the induced conjugation action on H/K it easily is seen that the action of G/H on B is regular (i.e. transitive and free). Hence it is readily veriﬁed that for each g ∈ T1, ψ(ge) is a permutation matrix, and

 1 1 ··· 1 1   1 1 ··· 1 1    1  1 1 ··· 1 1    ψ(Tc1e) =  . . . . .  . n  ......     1 1 ··· 1 1  1 1 ··· 1 1

56 2.2 primitive idempotents of QG

Clearly ψ(Tc1e) has eigenvalues 1 and 0, with respective eigenspaces

V1 = vect{(1, 1,..., 1)} and

V0 = vect{(1, −1, 0,..., 0), (1, 0, −1,..., 0),..., (1, 0, 0,..., −1)}, where vect(S) denotes the vector space generated by the set S. Hence

−1 ψ(Tc1e) = PE11P , where we denote by Eij ∈ Mn(F ) the elementary matrices whose entries are all 0 except in the (i, j)-spot, where it is 1. One knows that {E11,E22,...,Enn} and hence also

−1 −1 −1 {ψ(Tc1e) = PE11P ,PE22P ,...,PEnnP }

forms a complete set of orthogonal primitive idempotents of Mn(F ). Let y = −1 ψ(xe) = P AP . As −1 n−1 −n+1 E22 = AE11A ,...,Enn = A E11A we obtain that

−1 n−1 −n+1 {ψ(Tc1e), yψ(Tc1e)y , . . . , y ψ(Tc1e)y }

forms a complete set of orthogonal primitive idempotents of Mn(F ). Hence, applying ψ−1 gives us a complete set of orthogonal primitive idempotents of QGe. Next we will describe a complete set of matrix units in a simple component QGe(G, H, K) for a strong Shoda pair (H,K) of a ﬁnite group G. Corollary 2.2.2 Let (H,K) be a strong Shoda pair of a ﬁnite group G such that the twisting τ(gH, g0H) = 1 for all g, g0 ∈ N. We use the notation of Theorem 2.2.1 0 and for every x, x ∈ T2 hxei, let

0−1 Exx0 = xTc1εx .

0 Then {Exx0 : x, x ∈ T2 hxei} is a complete set of matrix units in QGe, i.e. e = P E and E E = δ E , for every x, y, z, w ∈ T hx i. x∈T2hxei xx xy zw yz xw 2 e Moreover ExxQGExx ' F , where F is the ﬁxed subﬁeld of QHε under the natural action of NG(K)/H.

57 wedderburn decomposition and idempotents

Proof This follows at once from Theorem 2.2.1 and the fact that QGe ' M[G:H](F ).

In order to obtain an internal description within the group algebra QG, −1 −1 one would like to write the element xe = ψ (P AP ) of Theorem 2.2.1 in terms of group ring elements of QG. It might be a hard problem to obtain a generic formula. One of the reasons is that we first need to describe a normal N/H basis of Q(ζ[H:K])/Q(ζ[H:K]) . In general this is difficult to do. However, one can find some partial results in the literature. For example Dirk Hachen- berger [Hac00] studied normal bases for cyclotomic fields Q(ζqm ) with q an odd prime number. Once this obstacle is overcome one can determine xe as P follows. Denote by ∆ : CN → C the trace map g∈N agg 7→ a1. It is easy to 1 1 P see and well known that ∆(α) = |N| χreg(α) = |N| χ∈Irr(N) χ(1)χ(α), where we denote by χreg the regular character of N and by Irr(N) the set of irre- P −1 ducible complex characters of N. It follows that xe = g∈N ∆(xeg )g = 1 P P −1 |N| g∈N χ∈Irr(N) χ(1)χ(xeg )g. Because ψ can be seen as the representation induced to N by a linear character of H with kernel K, ψ is an irreducible complex representation of N. As we know that xe belongs to a simple component of QN, namely the only one on which ψ does not vanish, and the primitive central idempotent of QN of such component is the sum of the primitive central idempotents of CN associated to the irreducible characters of the form σ◦T ◦ψ, with σ ∈ Gal(F/Q) and T is the map associating a matrix with its trace. We −1 deduce that χ(xeg ) vanishes in all the irreducible characters different from σ ◦T ◦ψ. Thus x = 1 P P (σ ◦T ◦ψ)(1)(σ ◦T ◦ψ)(x g−1)g = e |N| g∈N σ∈Gal(F/Q) e 1 P −1 −1 |H| g∈N trF/Q(T (P AP ψ(g )))g. We now show that we can sometimes also overcome the difficulties above using only basic linear algebra for the group C7 o1 C3, where the method from Theorem 1.7.19 failed. 7 3 b 2 Example 2.2.3 Let G = C7 o1 C3 = a, b : a = 1 = b , a = a . Then a GAP-computation shows that

2 4 QG = Q ⊕ Q(ζ3) ⊕ M3(Q(ζ7 + ζ7 + ζ7 )).

Consider the strong Shoda pair (hai , 1) that contributes to the unique non- 2 4 commutative simple component M3(Q(ζ7 +ζ7 +ζ7 )). The associated primitive central idempotent is e = e(G, hai , 1) = ε(hai , 1) and

2 4 M3(Q(ζ7 + ζ7 + ζ7 )) ' QGe ' Q hai e ∗ hbi

58 2.2 primitive idempotents of QG with trivial twisting. Consider the algebra isomorphism

2 4 ψ : Q hai e ∗ hbi ' M3(Q(ae + a e + a e)) with respect to B = {ae, a2e, a4e}, a normal basis of Q(ae) over its subﬁeld 2 4 Q(ae + a e + a e). Now we have A = ψ(be) and in order to describe xe = ψ−1(P AP −1) in terms of elements of QG, it is suﬃcient to write ψ−1(P ) in −1 2 terms of group ring elements. Write ψ (P ) = α0+α1b+α2b with αi ∈ Q hai e and solve the system of equations:

 0 0 0 2 2 4 (α0 + α1 ◦ b + α2 ◦ b )(ae) = (a + a + a )e  0 0 0 2 2 2 (α0 + α1 ◦ b + α2 ◦ b )(a e) = (a − a )e  0 0 0 2 4 4 (α0 + α1 ◦ b + α2 ◦ b )(a e) = (a − a )e.

2 3 4 5 This is done by writing each αi = (xi,0 +xi,1a+xi,2a +xi,3a +xi,4a +xi,5a )e 2 3 4 5 6 with xi,j ∈ Q and using the equality (1 + a + a + a + a + a + a )e = 0. This leads to a system of 18 linear equations in 18 variables. It can be veriﬁed that Å 4 1 1 5 1 5 ã ψ−1(P ) = − − a − a2 − a3 − a4 − a5 e 7 14 2 14 7 14 Å2 11 3 1 1 9 ã + − a − a2 − a3 − a4 − a5 be 7 14 14 2 7 14 Å2 9 11 9 2 1 ã + − a − a2 − a3 + a4 − a5 b2e. 7 14 14 14 7 2

−1 −2 2 By Theorem 2.2.1, the set {bbe, xe bbexe, xe bbexe} is a complete set of orthogonal primitive idempotents in QGe. This method of Theorem 2.2.1 yields a detailed description of a complete set of orthogonal primitive idempotents of QG when G is a ﬁnite strongly monomial group such that there exists a complete and non-redundant set of 0 0 strong Shoda pairs (H,K) satisfying τ(gH, g H) = 1 for all g, g ∈ NG(K). However, even when the group is not strongly monomial or some strong Shoda pairs yield a non-trivial twisting, our description of primitive idempotents can still be used in the components determined by a strong Shoda pair with trivial twisting. Nevertheless, it is crucial that the twistings appearing in the simple components are trivial in order to make use of Theorem 1.5.2. The following example shows that our methods cannot be extended to, for example, Cq o Cp2 with non-faithful action.

59 wedderburn decomposition and idempotents

Example 2.2.4 Consider the group

19 9 b 7 G = C19 o3 C9 = ha, b : a = b = 1, a = a i

and the strong Shoda pair ( a, b3 , 1). Let e be the associated primitive central idempotent e = e(G, a, b3 , 1) = ε( a, b3 , 1). The elements 1, b, b2 are coset representatives for a, b3 in G. Since b2 a, b3 b2 a, b3 = b a, b3 and b3 = 3 19 19 (ab ) , we get that τb2ha,b3i,b2ha,b3i = ζ57 6= 1. Hence the twisting is not trivial. We show now that our method applies to all metacyclic groups of the form Cqm o1 Cpn , for p and q different prime numbers. Recall that this notation means that Cpn acts faithfully on Cqm . Corollary 2.2.5 Let p and q be different prime numbers, m and n positive integers and m n G = hai o1 hbi with |a| = q and |b| = p . Each non-commutative simple component of QG is realized as QGε(hai ,Kj) for a strong Shoda pair ¨ qj ∂ (hai ,Kj) fore some 1 ≤ j ≤ m and where Kj = a . Furthermore, let Fj be the center of QGε(hai ,Kj), fix a normal element wj of Q(ζqj )/Fj and let Bj be the normal basis determined by wj. Let ψj : QGε(hai ,Kj) → Mpn (Fj) be the isomorphism given by The- orem 1.5.2 with respect to Bj. Then,

h −k n {xj bbxj : 1 ≤ h, k ≤ p }

is a complete set of matrix units of QGε(hai ,Kj), where

−1 −1 −1 xj = ψj (P )bε(hai ,Kj)ψj (P ) .

Proof The primitive central idempotents and simple components of QG are described in Corollary 1.7.18. Therefore, the ﬁrst statement follows immediately. ¨ qj ∂ Let Kj = a and consider the simple component QGε (hai ,Kj) ' Q ζqj ∗ Cpn .

Cpn It is easy to verify that the twisting is trivial. Let Fj = Q ζqj , the ﬁxed ﬁeld of Q ζqj by the action of Cpn . Let Bj be the normal basis determined by a normal element wj of Q(ζqj )/Fj. Let ψj : QGε(hai ,Kj) → Mpn (Fj) be the

60 2.3 primitive idempotents of FG

isomorphism given by Theorem 1.5.2 with respect to Bj. Then ψj(bε(hai ,Kj)) is the permutation matrix A of Theorem 2.2.1 and hbi is a transversal of hai in G. The result follows now from Corollary 2.2.2.

As we have shown, the groups of the form Cqm o1Cpn do satisfy the condition of a trivial twisting. However not all groups satisfying this condition on the twistings are metacyclic, for example the symmetric group S4 and the alternating group A4 of degree 4 have a trivial twisting in all Wedderburn components of their rational group rings and are not metacyclic (and not nilpotent). Triv- ially all abelian groups are included and it is also easy to prove that for all n 2 b −1 dihedral groups D2n = a, b : a = b = 1, a = a there exists a complete and non-redundant set of strong Shoda pairs with trivial twisting since the group action involved has order 2 and hence is faithful. On the other hand for 2n 4 n 2 y −1 quaternion groups Q4n = x, y : x = y = 1, x = y , x = x , one can verify that the strong Shoda pair (hxi , 1) yields a non-trivial twisting.

2.3 primitive idempotents of FG

In this section, we construct a complete set of orthogonal primitive idempotents in some semisimple finite group algebras FG. Apart from the ring theoretical interest, primitive idempotents can be used to construct linear codes. Let F be a finite field. A linear code over F of length n and rank k is a linear subspace C with dimension k of the vector space Fn. The standard basis of n F is denoted by E = {e1 , . . . , en }. The vectors in C are called codewords, the size of a code is the number of codewords and equals |F|k . If G is a group of order n and C ⊆ Fn is a linear code, then one says that C is a left G-code (respectively a G-code) if there is a bijection φ : E → G such that the linear extension of φ to an isomorphism φ : Fn → FG maps C to a left ideal (respectively a two-sided ideal) of FG.A left group code (respectively a group code) is a linear code which is a left G-code (respectively a G-code) for some group G. A (left) cyclic group code (respectively, abelian, metacyclic, nilpotent group code, . . . ) is a linear code which is a (left) G-code for some cyclic group (respectively, abelian, metacyclic, nilpotent group, . . . ) G. The underlying group is not uniquely determined by the code itself. That means that it is possible that a (left) non-abelian group code can also be realized as an abelian group code.

61 wedderburn decomposition and idempotents

We first state a result describing the (non-central) primitive idempotents of finite semisimple group algebras of nilpotent groups. We opted to not include the proof in this thesis because of its length and its technical behavior. The proof can be found in [OVG11, Theorem 3.3]. Theorem 2.3.1 (Olteanu-Van Gelder) Let F be a finite field and G a finite nilpotent group such that FG is semisimple. Let (H,K) be a strong Shoda pair of G, C ∈ CF(H/K) and set eC = eC (G, H, K), εC = εC (H,K), H/K = hai, E = EF(G, H/K). Let E2/K and H2/K = ha2i (respectively E20 /K and H20 /K = ha20 i) denote the 2-parts (respectively 20-parts) of E/K and H/K respectively. Then ha20 i has a cyclic complement hb20 i in E20 /K. A complete set of orthogonal primitive idempotents of FGeC consists of 0 0 the conjugates of βeC = bc2 β2εC by the elements of TeC = T2 T2TE, where 2 [E20 :H20 ]−1 T20 = {1, a20 , a20 , . . . , a20 }, TE denotes a right transversal of E in G and β2 and T2 are given according to the cases below.

1. If H2/K has a complement M2/K in E2/K then β2 = M”2. More- over, if M2/K is cyclic, then there exists b2 ∈ E2 such that E2/K is given by the following presentation

n k 2 2 b2 r ha2, b2 : a2 = b2 = 1, a2 = a2 i,

and if M2/K is not cyclic, then there exist b2, c2 ∈ E2 such that E2/K is given by the following presentation

n k 2 2 2 b2 r ha2, b2, c2 : a2 = b2 = c2 = 1, a2 = a2 ,

c2 −1 a2 = a2 , [b2, c2] = 1i,

2n−2 with r ≡ 1 mod 4 (or equivalently a2 is central in E2/K).

2 2k−1 2n−2 a) T2 = {1, a2, a2, . . . , a2 }, if a2 is central in E2/K (unless n ≤ 1) and M2/K is cyclic; and

n−2 2 d/2−1 2n−2 2n−2+1 2 +d/2−1 b) T2 = {1, a2, a2, . . . , a2 , a2 , a2 , . . . , a2 }, where d = [E2 : H2], otherwise.

62 2.3 primitive idempotents of FG

2. If H2/K does not have a complement in E2/K, then there exist b2, c2 ∈ E2 such that E2/K is given by the following presentation

2n 2k 2 2n−1 ha2, b2, c2 : a2 = b2 = 1, c2 = a2 ,

b2 r c2 −1 a2 = a2 a2 = a2 , [b2, c2] = 1i,

2n−2 2n−2 1+xa2 +ya2 c2 with r ≡ 1 mod 4. In this case, β2 = b“2 2 and

2 2k−1 2 2k−1 T2 = {1, a2, a2, . . . , a2 , c2, c2a2, c2a2, . . . , c2a2 },

with x, y ∈ F, satisfying x2 + y2 = −1 and y 6= 0.

Example 2.3.2 Consider once more the group G = C3 × Q8 from Exam- ples 2.1.9 and 2.1.14, the ﬁeld F5 and the strong Shoda pair (H,K), with H = ha, xi and K = 1. Take C = {1, 5}. We computed before that E =

0 0 EF5 (G, H/K) = H. Then E2 = H2 = hxi and E2 = H2 = hai. By The- orem 2.3.1, F5GeC (G, H, K) has as a complete set of orthogonal primitive idempotents the set −1 {εC (H,K), y εC (H,K)y}.

Theorem 2.3.1 provides a straightforward implementation in GAP. Neverthe- less, in case 2, there might occur some difficulties finding solutions for the equation x2 + y2 = −1 for x, y ∈ F and y 6= 0. However, it is possible to overcome this problem. Note that here F has to be of odd order pn. If p ≡ 1 mod 4, 2 then y = −1 has a solution in Fp ⊆ F. Half of the elements α of Fp satisfy p−1 the equation α 2 = −1. So we can pick an α ∈ Fp at random and check if the equality is satisfied. If not, repeat the process. When we have found such p−1 an α, then take y = α 4 and x = 0. 2 If 2 is a divisor of n, then y = −1 has a solution in Fp2 ⊆ F because 2 2 p −1 p ≡ 1 mod 4. Half of the elements β of Fp2 satisfy the equation β 2 = −1. p2−1 Pick a β ∈ Fp2 randomly. If the equality is satisfied, then take y = β 4 and x = 0. Now assume that p 6≡ 1 mod 4 and 2 - n. Recall that the Legendre symbol (a/p) for an integer a and an odd prime number p, is defined as 1 if the congruence x2 ≡ a mod p has a solution, as 0 if p divides a and as −1 otherwise. Using the Legendre symbol, one can decide whether an element is a square

63 wedderburn decomposition and idempotents

modulo p or not and this can be eﬀectively calculated using the properties of the Jacobi symbol as explained in standard references as, for example, in the book of Kenneth Ireland and Michael Rosen [IR90]. Take now a random element a ∈ Fp ⊆ F and check if both a and −1 − a are squares in Fp. If so, then one can use the algorithm of Tonelli and Shanks or the algorithm of Cornacchia to compute square roots modulo p and to 2 2 ﬁnd x and y in Fp ⊆ F satisfying x + y = −1. For more information on these algorithms the reader is referred to the literature, for example [Coh93, Algorithms 1.5.1 and 1.5.2]. We illustrate this in an example. Example 2.3.3 Consider the group with the following presentation

G = a, b : a4 = 1, b12 = 1, b−1ab = a−1

and its strong Shoda pair (H,K) with H = a, b2 and K = a2b6 . Then 2 H/K = ab K ' C12 and I12(F7) ' Gal(F7(ζ12)/F7) = Gal(F49/F7) ' −1 2 2 7 {1, 7}. Then EF7 (G, H/K) = G since b ab bK = (ab ) K and 7 ∈ I12(F7). Take C = {1, 7} ∈ CF7 (G, H/K). We will compute a complete set of orthogonal primitive idempotents in F7GeC (G, H, K). 3 Following Theorem 2.3.1, we compute E2 = G2 = a, b , E20 = G20 = 4 6 b = H20 and H2 = a, b . Then H2/K does not have a complement in G2/K and we are in case 2. We ﬁnd the following presentation for G2/K:

hab6K, b3K :(ab6K)4 = 1, (b3K)2 = (ab6K)2, (b3K)−1(ab6K)(b3K) = (ab6K)−1i.

Since 2 ≡ 32 mod 7 and −3 ≡ 4 ≡ 52 mod 7, both 2 and −3 are squares in 2 2 F7 and hence we find that x = 3 and y = 5 satisfy the equation x + y = −1 3 1+3ab6+5ab9 b3 in F7. We define T2 = {1, b } and β2 = 2 . Now the set {β2, β2 } is a complete set of orthogonal primitive idempotents in F7GeC (G, H, K). Computations involving strong Shoda pairs and primitive central idempotents were already provided in the GAP package wedderga [BHK+14] and we have included the function PrimitiveIdempotentsNilpotent implementing the above algorithm. Assume now that F is a finite field of order q and G is a finite group such that the order of G is coprime to q. We give an analogue to Theorem 2.2.1 and focus on simple components of FG which are determined by a strong 0 Shoda pair (H,K) and a class C ∈ CF(H/K) such that τ(gH, g H) = 1 for all

64 2.3 primitive idempotents of FG

0 g, g ∈ E = EF(G, H/K). For such a component, we describe a complete set of orthogonal primitive idempotents. Before we do so, we need a normal element w ∈ F(ζ[H:K]) and a normal basis {wx : x ∈ E/H} of

E/H F(ζ[H:K])/F(ζ[H:K]) = Fqo /Fqo/[E:H]

(with o the multiplicative order of q modulo [H : K]). Recall that E/H, the Galois group of Fqo over Fqo/[E:H] , is cyclic and generated by the Frobenius qo/[E:H] automorphism x 7→ x . Hence if w ∈ Fqo is such that the [E : H] elements o/[E:H] o/[E:H] [E:H]−1 {w, wq , . . . , w(q ) }

are linearly independent, then this set forms a normal basis for Fqo over Fqo/[E:H] . The existence of such a basis is stated in the Normal Basis The- orem 1.3.1. For background on the construction of normal bases, see Emil Artin [Art44], Heinz Lüneburg[Lün86],Hendrik W. Jr. Lenstra [Len91] and Shuhong Gao [Gao93]. The construction of normal bases for finite fields is also implemented in GAP in the method NormalBase. The proof of the following theorem is very similar to the one of Theorem 2.2.1 and is therefore omitted. Theorem 2.3.4 Let G be a finite group and F a finite field of order q such that q is coprime to the order of G. Let (H,K) be a strong Shoda pair of G such that 0 0 τ(gH, g H) = 1 for all g, g ∈ E = EF(G, H/K), and let C ∈ CF(H/K). Let ε = εC (H,K) and e = eC (G, H, K). Let w be a normal element of Fqo /Fqo/[E:H] (with o the multiplicative order of q modulo [H : K]) and B the normal basis determined by w. Let ψ be the isomorphism between FEε and the matrix algebra M[E:H](Fqo/[E:H] ) with respect to the basis B as stated in Theorem 1.5.2. Let P,A ∈ M[E:H](Fqo/[E:H] ) be defined as follows:

 1 1 1 ··· 1 1   0 0 ··· 0 1   1 −1 0 ··· 0 0   1 0 ··· 0 0       1 0 −1 ··· 0 0   0 1 ··· 0 0  P =  ......  and A =  . . . . .  .  ......   . . . . .   ......   . . . . .   1 0 0 · · · −1 0   0 0 ··· 0 0  1 0 0 ··· 0 −1 0 0 ··· 1 0

65 wedderburn decomposition and idempotents

Then −1 {xTc1εx : x ∈ T2 hxei}

is a complete set of orthogonal primitive idempotents of FGe where xe = −1 −1 ψ (P AP ), T1 is a transversal of H in E and T2 is a right transversal 1 P of E in G. By T1 we denote the element t in G. c |T1| t∈T1 F

We have included an implementation of the above theorem in the function PrimitiveIdempotentsTrivialTwisting in wedderga. This was possible because GAP can easily ﬁnd a normal basis and we can compute ψ−1(P AP −1) algorithmically. Using our implementation in wedderga of primitive idempotents of ﬁnite semisimple group algebras described in Theorems 2.3.1 and 2.3.4, it is possible to construct many left group codes. For more details and examples on this topic, the reader is referred to our joined work with Gabriela Olteanu and the references given in [OVG15].

2.4 conclusions

We made some progress on the description of non-central primitive idempotents of rational group algebras QG and of finite semisimple group algebras FG. However, there are still cases to study, for example to cover all classes of non-nilpotent groups within the strongly monomial groups or to replace the field Q or F with a number field F . Also for the description of primitive central idempotents one is somehow limited to strongly monomial groups. The ultimate goal would be to study idempotents in rational group algebras of all finite groups, including simple groups. Since all our constructions are based on the idempotent N“, for a non-trivial normal subgroup N, none of our methods apply to simple groups. However, some recent results (see [JOdR12, Jan13, Olt07]) show that the computation of the primitive central idempotents of QG for G a finite arbitrary group can be reduced to the case of finite strongly monomial groups, which means that this case is essential for the computation of both the primitive central idempotents and the Wedderburn decomposition of a rational group algebra.

66 3

EXCEPTIONALCOMPONENTS

When studying the group of units of RG for a finite group G and the ring of integers R of a number field F , one is often restricted to groups such that no exceptional component appears in the Wedderburn decomposition of FG.A good example for this is Proposition 1.13.3. Therefore, it is useful to classify finite groups G and abelian number fields F such that FG contains an exceptional component in its Wedderburn decomposition. Recall from Corollary 1.9.9, that we know exactly which isomorphism types of exceptional components can occur. For the unit groups of exceptional components of type EC1 very little is known. Let O be a Z-order in a non-commutative division ring different from a totally definite quaternion algebra. Then SL1(O) is infinite and to the best of our knowledge, there are no generic constructions of subgroups of finite index known. In 2000, Kleinert [Kle00b] gave a commendable survey on that topic. Up to that date no constructions were known for degree 3 division rings and also for degree 2 very little was known. Only recently, there was some progress made by [CJLdR04, JJK+15] for degree 2 division rings. Braun, Coulangeon, Nebe and Schönnenbeck provided a generalization of Vorono¨ı’salgorithm to tackle the problem [BCNS], they give examples to work with division algebras of degree 2 and 3. Still one would like to have generic constructions, and moreover to have constructions of groups of units in RG that contain a subgroup of finite index in U(O), when O is a Z-order in a division ring appearing in the Wedderburn decomposition of FG. In this chapter we first classify all exceptional components of type EC2 occurring in the Wedderburn decomposition of group algebras of finite groups over arbitrary number fields. We do this by giving a full list of finite groups G, number fields F and exceptional components M2(D) such that M2(D) is a faithful Wedderburn component of FG, cf. Theorem 3.1.2.

67 exceptional components

Afterwards we deal with exceptional components of type EC1. We classify F -critical groups, i.e. groups G such that FG has an exceptional component of type EC1 in its Wedderburn decomposition, but no proper quotient has this property. In this way we obtain a minimal list of exceptional components of type EC1 appearing in group algebras FG for abelian number fields F and G finite. Regarding the scale of the difficulty of the problem, at least for the division rings in the list the unit groups of Z-orders have to be studied. Note that any group H such that FH has a non-commutative division ring (not totally definite quaternion) in its Wedderburn decomposition has an epimorphic F -critical image G such that if an exceptional component D of type EC1 appears as a faithful Wedderburn component of FG, then also FH has D as a simple component. Having an F -critical epimorphic image for a group implies that, up to now, there is no hope for a generic construction of units in RG. We give necessary and sufficient conditions for a finite group G to be F -critical expressed in easy arithmetic formulas in terms of parameters of the group, and ramification indices and residue degrees of extensions of F , rely- ing on the parameters of the group. For any abelian number field F and any finite F -critical group G, we explicitly describe the division ring, which is an exceptional component of type EC1 in FG, cf. Theorem 3.2.21. All results presented in this chapter extend a result of Mauricio Caicedo and Angel´ del R´ıo[CdR14] where they handled the case F = Q. The results of Theorems 1.9.6, 1.9.8 and 1.10.3 and Proposition 1.9.7 give more insight in the exceptional components of type EC2, therefore it is natural to distinguish between exceptional components of type EC1 and type EC2, in contrast to what was done in [CdR14]. So our work yields (also for the case of rational group algebras) an extension of the above cited result.

3.1 group algebras with exceptional components of type ec2

We call a Wedderburn component A of FG faithful if G is faithfully embedded in A via the Wedderburn isomorphism. In this section, we give a full list of finite groups G and number fields F having faithful exceptional components of type EC2 in FG. Employing Lemma 3.1.1 one can deduce from this list all exceptional components of type EC2 within group algebras over number fields.

68 3.1 group algebras with exceptional components of type ec2

Lemma 3.1.1 Let G be a ﬁnite group, F be a number ﬁeld and ρ an irreducible F - representation of G. Let e be the primitive central idempotent associated to ρ and K be the kernel of ρ. Then the group Ge is faithfully embedded in (FG)e and its F -span equals (FG)e. In particular, Ge is isomorphic to a subgroup of U((FG)e).

Proof Consider the irreducible representation ρ: G → U(F Ge). This induces a faithful representation ρ: G/K → U(F Ge) and G/K ' ρ(G) = Ge. Since F Ge is the F -span of ρ(G) = ρ(G/K) and F Ge is simple, F Ge is also isomorphic to a Wedderburn component of F (G/K). Theorem 3.1.2 Let F be a number ﬁeld, G be a ﬁnite group and B a simple exceptional algebra of type EC2. Then B is a faithful Wedderburn component of FG if and only if G, F , B is a row listed in Table 2.

Proof Let B be a simple exceptional algebra of type EC2 and assume that B is a faithful Wedderburn component of FG, then by Lemma 3.1.1, G is a subgroup of U(B) and B is isomorphic to an algebra stated in Theorems 1.9.6 and 1.9.8. √ √ √ The subgroups of M2(Q),M2(Q( −1)),M2(Q( −2)),M2(Q( −3)) are embedded in GL(2, 25) by Proposition 1.9.7. The maximal finite subgroups of 2×2-matrices over totally definite quaternion algebras with center Q were classified in Theorem 1.10.3. They can be accessed in Magma [BCP97] by calling QuaternionicMatrixGroupDatabase(). It is also clear that when FG has a Wedderburn component B then F is contained in the center of B, which restricts the possibilities of F for G and B fixed. Additionally, using the GAP-package wedderga, one can compute a finite list of groups G that have B as a faithful component over F . We mainly use the function WedderburnDecompositionWithDivAlgParts which returns the size of the matrices, the centers and the local indices of all Wedderburn components of a group algebra and allows us to compare the Wedderburn components to the possibilities of B above. This is possible, since F is a number field and the isomorphism type of division algebras is determined by its list of local Hasse invariants at all primes of F (Theorem 1.4.2). For quaternion algebras the local Hasse invariants are uniquely determined by the local Schur indices.

69 exceptional components

Notation 3.1.3 (in Table 2) We use the GAP notation for the group structure and the identiﬁcation number from the SmallGroups library. For a non- split extension of A by B, we write A.B. If an exceptional component appears several times in FG, this multiplicity is indicated in the last column.

Table 2: List of all groups having a faithful exceptional component of type EC2 ID Structure FB

[6, 1] S3 Q 1× M2 (Q) √ √ [6, 1] S3 Q( −1) 1× M2 Q( −1) √ √ [6, 1] S3 Q( −2) 1× M2 Q( −2) √ √ [6, 1] S3 Q( −3) 1× M2 Q( −3) [8, 3] D8 Q 1× M2 (Q) √ √ [8, 3] D8 Q( −1) 1× M2 Q( −1) √ √ [8, 3] D8 Q( −2) 1× M2 Q( −2) √ √ [8, 3] D8 Q( −3) 1× M2 Q( −3) √ √ [8, 4] Q8 Q( −1) 1× M2 Q( −1) √ √ [8, 4] Q8 Q( −2) 1× M2 Q( −2) √ √ [8, 4] Q8 Q( −3) 1× M2 Q( −3) √ √ [12, 1] C3 o C4 Q( −1) 1× M2 Q( −1) √ √ [12, 1] C3 o C4 Q( −3) 1× M2 Q( −3) [12, 4] D 1× M ( ) 12 Q √ 2 Q √ [12, 4] D ( −1) 1× M ( −1) 12 Q √ 2 Q √ [12, 4] D ( −2) 1× M ( −2) 12 Q √ 2 Q √ [12, 4] D ( −3) 1× M ( −3) 12 Q 2 Q √ [16, 6] C C 1× M ( −1) 8 o 2 Q √ 2 Q √ [16, 6] C C ( −1) 2× M ( −1) 8 o 2 Q 2 Q √ [16, 8] QD (also denoted by D− ) 1× M ( −2) 16 16 Q √ 2 Q √ − [16, 8] QD16 (also denoted by D16) Q( −2) 2× M2 Q( −2) √ [16, 13] (C4 × C2) o C2 Q 1× M2 Q( −1) √ √ [16, 13] (C4 × C2) o C2 Q( −1) 2× M2 Q( −1) √ [18, 3] C3 × S3 Q 1× M2 Q( −3) √ √ [18, 3] C3 × S3 Q( −3) 2× M2 Q( −3) √ [24, 1] C3 o C8 Q 1× M2 Q( −1) √ √ [24, 1] C3 o C8 Q( −1) 2× M2 Q( −1) √ [24, 3] SL(2, 3) Q 1× M2 Q( −3) √ √ [24, 3] SL(2, 3) Q( −1) 1× M2 Q( −1) √ √ [24, 3] SL(2, 3) Q( −2) 1× M2 Q( −2) continued

70 3.1 group algebras with exceptional components of type ec2

ID Structure FB √ √ [24, 3] SL(2, 3) Q( −3) 3× M2 Q( −3) √ [24, 5] C4 × S3 Q 1× M2 Q( −1) √ √ [24, 5] C4 × S3 Q( −1) 2× M2 Q( −1) √ [24, 8] (C6 × C2) o C2 Q 1× M2 Q( −3) √ √ [24, 8] (C6 × C2) o C2 Q( −3) 2× M2 Q( −3) √ [24, 10] C3 × D8 Q 1× M2 Q( −3) √ √ [24, 10] C3 × D8 Q( −3) 2× M2 Q( −3) √ [24, 11] C3 × Q8 Q 1× M2 Q( −3) √ √ [24, 11] C3 × Q8 Q( −3) 2× M2 Q( −3) −1,−1 [32, 8] (C2 × C2).(C4 × C2) Q 1× M2 Q√ [32, 11] (C4 × C4) o C2 Q 2× M2 Q( −1) √ √ [32, 11] (C4 × C4) o C2 Q( −1) 4× M2 Q( −1) −1,−1 [32, 44] (C2 × Q8) C2 1× M2 o Q Q −1,−1 [32, 50] (C2 × Q8) o C2 Q 1× M2 Q√ [36, 6] C3 × (C3 o C4) Q 1× M2 Q( −3) √ √ [36, 6] C3 × (C3 o C4) Q( −3) 2× M2 Q( −3) √ [36, 12] C6 × S3 Q 1× M2 Q( −3) √ √ [36, 12] C6 × S3 Q( −3) 2× M2 Q( −3) −2,−5 [40, 3] C5 C8 1× M2 o Q Q −1,−1 [48, 16] (C3 C8) C2 1× M2 o o Q Q −1,−3 [48, 18] C3 Q16 1× M2 o Q Q ∗ −1,−3 [48, 28] SL(2, 3).C2 = O Q 1× M2 Q√ [48, 29] GL(2, 3) Q 1× M2 Q( −2) √ √ [48, 29] GL(2, 3) Q( −2) 2× M2 Q( −2) √ [48, 33] SL(2, 3) o C2 Q 1× M2 Q( −1) √ √ [48, 33] SL(2, 3) o C2 Q( −1) 2× M2 Q( −1) −1,−3 [48, 39] (C2 × (C3 C4)) C2 1× M2 o o Q Q −1,−1 [48, 40] Q8 × S3 1× M2 Q Q −1,−1 [64, 37] (C4 × C2).(C4 × C2) 2× M2 Q Q −1,−1 [64, 137] ((C4 × C4) C2) C2 2× M2 o o Q Q −1,−3 [72, 19] (C3 × C3) C8 2× M2 o Q Q −1,−3 [72, 20] (C3 C4) × S3 1× M2 o Q Q −1,−3 [72, 22] (C6 × S3) C2 1× M2 o Q Q −1,−3 [72, 24] (C3 × C3) o Q8 Q 1× M2 Q√ [72, 25] C3 × SL(2, 3) Q 3× M2 Q( −3) √ √ [72, 25] C3 × SL(2, 3) Q( −3) 6× M2 Q( −3) √ [72, 30] C3 × ((C6 × C2) o C2) Q 2× M2 Q( −3) continued

71 exceptional components

ID Structure FB √ √ [72, 30] C3 × ((C6 × C2) o C2) Q( −3) 4× M2 Q( −3) √ [96, 67] SL(2, 3) o C4 Q 2× M2 Q( −1) √ √ [96, 67] SL(2, 3) o C4 Q( −1) 4× M2 Q( −1) −1,−1 [96, 190] (C2 × SL(2, 3)) C2 1× M2 o Q Q ∗ −1,−1 [96, 191] (SL(2, 3).C2 = O ) C2 1× M2 o Q Q −1,−1 [96, 202] (C2 × SL(2, 3)) C2 1× M2 o Q Q −1,−3 [120, 5] SL(2, 5) 1× M2 Q Q −1,−1 [128, 937] (Q8 × Q8) C2 4× M2 o Q Q ∗ −1,−3 [144, 124] C3 (SL(2, 3).C2 = O ) 3× M2 o Q Q −1,−1 [144, 128] S3 × SL(2, 3) 1× M2 Q Q −1,−3 [144, 135] ((C3 × C3) C8) C2 4× M2 o o Q Q −1,−3 [144, 148] (C2 × ((C3 × C3) C4)) C2 2× M2 o o Q Q −1,−1 [160, 199] ((C2 × Q8) C2) C5 1× M2 o o Q Q ∗ −1,−1 [192, 989] ((SL(2, 3).C2 = O ) C2) C2 2× M2 o o Q Q −2,−5 [240, 89] C2.S5 = SL(2, 5).C2 1× M2 Q Q −2,−5 [240, 90] SL(2, 5) C2 1× M2 o Q Q −1,−3 [288, 389] ((C3 C4) × (C3 C4)) C2 2× M2 o o o Q Q −1,−1 [320, 1581] (((C2 × Q8) C2) C5).C2 2× M2 o o Q Q −1,−1 [384, 618] ((Q8 × Q8) C3) C2 1× M2 o o Q Q −1,−1 [384, 18130] ((Q8 × Q8) C3) C2 1× M2 o o Q Q −1,−3 [720, 409] SL(2, 9) 2× M2 Q Q −1,−1 [1152, 155468] (SL(2, 3) × SL(2, 3)) C2 1× M2 o Q Q −1,−1 [1920, 241003] C2.((C2 × C2 × C2 × C2) A5) 1× M2 o Q Q

3.2 group algebras with exceptional components of type ec1

In this section we consider exceptional components of type EC1. We provide necessary and sufficient conditions for a finite group G to be F -critical. Let G be a finite group and F an abelian number field. We say that G is F -critical if and only if

1. FG has a Wedderburn component which is exceptional of type EC1, and

2. for any 1 6= N ¢ G the group algebra F (G/N) does not have a Wedder- burn component which is exceptional of type EC1.

72 3.2 group algebras with exceptional components of type ec1

Note that if a group G is F -critical with corresponding exceptional component B, the F -representation of G associated to B is necessarily faithful. In particular G is in the classiﬁcation of Amitsur (cf. Theorem 1.10.1).

Lemma 3.2.1 A division algebra A is exceptional if and only if ind(A) > 2 or ind(A) = 2 and m∞(A) 6= 2.

Proof This follows immediately since a quaternion algebra is totally deﬁnite if and only if its local index at inﬁnity is 2.

We recall that for a ﬁnite Galois extension F/Q, always F is either totally real or totally imaginary.

Proposition 3.2.2 Ä a,b ä Let A = K with K a totally real finite Galois extension of Q, a, b ∈ K totally negative and let F be a finite Galois extension of Q containing K. Then F ⊗K A is a division algebra if and only if F is a totally real number field or there exists a prime number p such that mp(A) 6= 1 and both ep(F/K) and fp(F/K) are odd.

Proof Let B = F ⊗K A. Note that B is a division algebra if and only if B is not split, as the degree is 2. Furthermore, B is not split if and only if mp(B) 6= 1 for some prime p of Q (finite or infinite). Assume that F is totally real, then there exists a real embedding σ of F Ä σ(a),σ(b) ä and for the completion of this embedding we find ⊗F B ' , hence R R m∞(B) = 2 and B is a division algebra. Assume that F is totally imaginary, then for all embeddings of F all local Schur indices for infinite primes are 1, since the completion of any embedding of F is C and splits B. Fix a finite prime number p. If mp(A) = 1, then clearly mp(B) = 1. If mp(A) 6= 1, then it is 2 and due to Theorem 1.4.8, mp(B) = 1 if and only if mp(A) | [Ffp : K›p]. Hence the result follows since [Ffp : K›p] = ep(F/K)fp(F/K) by Proposition 1.4.6.

Proposition 3.2.2 is a generalization of the following statement from [SW86, Theorem 2.1.9].

73 exceptional components

Corollary 3.2.3 Ä ä For F a ﬁnite Galois extension of , F ⊗ −1,−1 is a division algebra Q Q Q if and only if F is a totally real ﬁeld or both e2(F/Q) and f2(F/Q) are odd.

Ä −1,−1 ä Proof Let A = . Then m2(A) = 2 and mp(A) = 1 for all odd Q prime numbers p, see Example 1.4.12. The result easily follows from Proposi- tion 3.2.2. We first consider the NZ-groups from Amitsur’s classification in Proposi- tions 3.2.4 to 3.2.7 and 3.2.9. Proposition 3.2.4 Let F be an abelian number field and let O∗ be the binary octahedral group. Then O∗ is never F -critical.

Proof The Wedderburn decomposition of QO∗ equals Ç−1, −1å Å−1, −3ã 2Q ⊕ M2(Q) ⊕ 2M3(Q) ⊕ √ ⊕ M2 . Q( 2) Q Hence the only possible exceptional component of type EC1 of F O∗ can come from √ [F ∩ ( 2): ] Ç−1, −1å Ç−1, −1å Q Q F ⊗Q √ = √ . Q( 2) F ( 2) The quaternion algebra −1√,−1 has all local Schur indices 1, except the local Q( 2) −1,−1 −1,−1 index m∞ √ = 2. By Proposition 3.2.2, √ is therefore only a Q( 2) F ( 2) division algebra when F is totally real. But in this case it is a totally deﬁnite quaternion algebra and hence not exceptional. Proposition 3.2.5 Let F be an abelian number ﬁeld. Then SL(2, 5) is never F -critical.

Proof The Wedderburn decomposition of QSL(2, 5) equals Ç−1, −1å Å−1, −3ã Q ⊕ M4(Q) ⊕ √ ⊕ M2 Q( 5) Q Å−1, −1ã √ ⊕ M5(Q) ⊕ M3 ⊕ M3(Q( 5)). Q

74 3.2 group algebras with exceptional components of type ec1

The only possible exceptional component of type EC1 of F SL(2, 5) can come from √ [F ∩ ( 5): ] Ç−1, −1å Ç−1, −1å Q Q F ⊗Q √ = √ . Q( 5) F ( 5) The algebra −1√,−1 has all local Schur indices 1, except m −1√,−1 = 2. ( 5) ∞ ( 5) Q Q By Proposition 3.2.2, −1√,−1 is therefore only a division algebra when F is F ( 5) totally real. But in this case it is a totally deﬁnite quaternion algebra and hence not exceptional. Proposition 3.2.6

Let F be an abelian number ﬁeld and Q4k be the generalized quaternion group with k even. Then Q4k is F -critical if and only if k = 2, F is totally imaginary and both e2(F/Q) and f2(F/Q) are odd. In this case FQ8 −1,−1 contains an exceptional component F .

2k k 2 −1 −1 t 0 Proof Let Q4k = a, b : a = 1, a = b , b ab = a with k = 2 k , t ≥ 1 0 and 2 - k , then by Lemma 1.7.17, the non-commutative components of QQ4k come from the strong Shoda pairs (hai , ad ) with d | 2k such that d 6= 1, 2. The corresponding simple components are

d QQ4ke(Q4k, hai , a ) ( −1 −1 (Q(ζd)/Q(ζd + ζd ), 1) = M2(Q(ζd + ζd )), if d | k = −1 −1,−1 (10) (Q(ζd)/Q(ζd + ζd ), −1) = −1 , otherwise. Q(ζd+ζd )

0 0 We claim that k = 1. For this, suppose that Q4k is F -critical and k 6= 1. By the above paragraph the exceptional component of FQ4k comes from

[F ∩ (ζ +ζ−1): ] Ç −1, −1 å Ç −1, −1 å Q d d Q F ⊗Q −1 = −1 Q(ζd + ζd ) F (ζd + ζd ) −1 Å Å ãã[F ∩Q(ζd+ζd ):Q] −1 −1, −1 = F (ζd + ζd ) ⊗ Q Q for some d satisfying that d | 2k and d - k. Note that this implies that t+1 0 0 0 0 d = 2 k1 where k1 | k and 2 - k1. This quaternion algebra cannot be totally deﬁnite (i.e. F is not totally real) although it is a division algebra. −1 −1 Hence, by Corollary 3.2.3, e2(F (ζd + ζd )/Q) and f2(F (ζd + ζd )/Q) are

75 exceptional components

0 odd. Due to k 6= 1, we have that Q t is a non-abelian proper quotient of Å ã 4·2 −1,−1 Q4k, moreover −1 is a simple component of FQ4·2t . Hence it F (ζ t+1 +ζ ) 2 2t+1 must be either a totally deﬁnite quaternion algebra or a 2 × 2-matrix ring Å ã −1 −1,−1 over F (ζ2t+1 + ζ t+1 ). If F splits the quaternion algebra −1 , by 2 (ζ t+1 +ζ ) Q 2 2t+1 −1 −1 Corollary 3.2.3, e2(F (ζ2t+1 +ζ2t+1 )/Q) or f2(F (ζ2t+1 +ζ2t+1 )/Q) is even. Since,

−1 e2(F (ζd + ζd )/Q) −1 −1 −1 = e2(F (ζd + ζd )/F (ζ2t+1 + ζ2t+1 ))e2(F (ζ2t+1 + ζ2t+1 )/Q) and

−1 f2(F (ζd + ζd )/Q) −1 −1 −1 = f2(F (ζd + ζd )/F (ζ2t+1 + ζ2t+1 ))f2(F (ζ2t+1 + ζ2t+1 )/Q), −1 −1 this implies that e2(F (ζd + ζ )/Q) or f2(F (ζd + ζ )/Q) is even, a contra- Å d ã d −1,−1 diction. Therefore −1 is a totally definite quaternion algebra, F (ζ t+1 +ζ ) 2 2t+1 −1 and then F (ζ2t+1 + ζ t+1 ) is totally real, and hence F is totally real, again a 2 Å ã 0 −1,−1 contradiction. Therefore k = 1 and, by formula (10), −1 is the F (ζ t+1 +ζ ) 2 2t+1 only candidate for being an exceptional component of type EC1. Å ã −1,−1 We claim that t = 1. By Corollary 3.2.3, −1 is a division F (ζ t+1 +ζ ) 2 2t+1 −1 −1 algebra whenever e2(F (ζ2t+1 +ζ2t+1 )/Q) and f2(F (ζ2t+1 +ζ2t+1 )/Q) are odd or −1 −1 F is totally real. Assume that t > 1, then F (ζ8 + ζ8 ) ⊆ F (ζ2t+1 + ζ2t+1 ) and −1 √ −1 e2(F (ζ8 + ζ8 )/Q) = e2(Q( 2)/Q) = 2 and hence also e2(F (ζ2t+1 + ζ t+1 )/Q) Å ã 2 −1,−1 is even. But in this case −1 is a totally definite quaternion F (ζ t+1 +ζ ) 2 2t+1 algebra and hence not exceptional. We conclude that t must be equal to 1 −1,−1 and k = 2. Since G is F -critical, F is not a totally definite quaternion algebra. Hence F is totally imaginary and e2(F/Q) and f2(F/Q) are odd. Finally we prove the converse. By the assumptions and Corollary 3.2.3, we −1,−1 have that F is an exceptional component of type EC1 of FQ8. The only proper quotients of Q8 are abelian groups and hence Q8 is F -critical. As we have seen, the reciprocity rules from Theorem 1.4.4 are very useful and often lead to an answer in a few steps. We also get an advantage of the use of those rules in the remaining cases (NZ)(d) and (NZ)(e).

76 3.2 group algebras with exceptional components of type ec1

Proposition 3.2.7 Let F be an abelian number ﬁeld.

1. SL(2, 3) is F -critical if and only if F is totally imaginary and both e2(F/Q) and f2(F/Q) are odd. In this case, F SL(2, 3) contains an −1,−1 exceptional component F . 2. Let M be a group in (Z) of order coprime to 6, such that 2 has odd order modulo |M|. Then SL(2, 3) × M is F -critical if and only if M is a cyclic group of prime order p, F is totally real and both e2(F (ζp)/Q) and f2(F (ζp)/Q) are odd. In this case, F (SL(2, 3)×Cp) Ä ä contains an exceptional component −1,−1 . F (ζp)

Proof 1. Let G = SL(2, 3). We ﬁrst assume that G is F -critical. The Wedderburn decomposition of QG equals Å−1, −1ã QG ' Q ⊕ Q(ζ3) ⊕ M3(Q) ⊕ ⊕ M2(Q(ζ3)). Q Ä ä The only exceptional component of type EC1 of FG can be F ⊗ −1,−1 = Q Q −1,−1 F , hence it is an exceptional component of FG. Using Corollary 3.2.3 the result follows. −1,−1 Now we prove the converse. By Corollary 3.2.3 we have that F is an exceptional component of FG of type EC1. On the other hand, G has only one non-abelian proper quotient which is isomorphic to A4, and the Wedderburn decomposition of QA4 equals

QA4 ' Q ⊕ Q(ζ3) ⊕ M3(Q). Hence the group algebra of any non-abelian proper quotient of G does not have exceptional components. So we conclude that G is F -critical. 2. Let G = SL(2, 3)×M. We first assume that G is F -critical. Observe that in the Wedderburn decomposition of FM totally definite quaternion algebras can not appear, since the order of M is odd. Moreover, due to the fact that M is a proper quotient of G, FM does not have non-commutative division algebras as simple components. Another non-abelian quotient of G is SL(2, 3), −1,−1 and according to the Wedderburn decomposition of QSL(2, 3), F has to be either a totally definite quaternion algebra or a 2 × 2-matrix ring over −1,−1 F . Suppose that F is a 2 × 2-matrix ring over F . The fact that

77 exceptional components

FG ' F SL(2, 3) ⊗F FM implies that there is not any division algebra in the Wedderburn decomposition of FG which is a contradiction. Therefore, −1,−1 F is a totally definite quaternion algebra and F is a totally real field. On the other hand, let D be an exceptional component of type EC1 of FG, then D ' D1 ⊗F D2 where D1 and D2 are simple components of F SL(2, 3) and FM respectively. Having in mind the Wedderburn decompositions of F SL(2, 3) and FM, and since D is a division algebra which is not a totally −1,−1 definite quaternion algebra, we can deduce that D1 ' F and D2 ' F (ζd) for some divisor d of the order of M, d 6= 1. We know that F (ζd) is a simple component of F (M/M 0) (and so D is a simple component of F (G/M 0)), hence by hypothesis M 0 is trivial, so that M is abelian and by the conditions in Theorem 1.10.1, M is a cyclic group. Now we claim that M has prime order. −1,−1 Let d be a proper divisor of |M|, then F ⊗F F (ζd) is a simple component of F (G/C|M|/d). By hypothesis it must be a 2 × 2-matrix ring over F (ζd), −1,−1 and it follows that D ' F ⊗F F (ζ|M|). By Corollary 3.2.3, F (ζd) is a totally imaginary field and e2(F (ζd)/Q) or f2(F (ζd)/Q) is even, moreover both e2(F (ζ|M|)/Q) and f2(F (ζ|M|)/Q) are odd. By Theorem 1.4.4, we have

e2(F (ζ|M|)/Q) = e2(F (ζ|M|)/F (ζd))e2(F (ζd)/Q) and f2(F (ζ|M|)/Q) = f2(F (ζ|M|)/F (ζd))f2(F (ζd)/Q) are both odd, but this is a contradiction since e2(F (ζd)/Q) or f2(F (ζd)/Q) is even. So the claim follows. By the above paragraph, we have that G = SL(2, 3) × Cp and Å−1, −1ã D ' ⊗ F (ζ ) F F p is an exceptional component of type EC1 of FG. Then again by Corollary 3.2.3 both e2(F (ζp)/Q) and f2(F (ζp)/Q) are odd. Now suppose that G = SL(2, 3) × Cp. Using Corollary 3.2.3 and having in mind the Wedderburn decomposition of QSL(2, 3), we can deduce that D ' −1,−1 F ⊗F F (ζp) is the unique exceptional component of FG of type EC1. Note that the non-abelian proper quotients of G are SL(2, 3), A4 and A4 × Cp. Due to the fact that F is a totally real field, F SL(2, 3) does not have exceptional components of type EC1. As in the Wedderburn decomposition of QA4 and Q(A4 × Cp) only fields and matrix rings show up, we have that FA4 and F (A4 × Cp) do not have division algebras as simple components. This finishes the proof.

78 3.2 group algebras with exceptional components of type ec1

Remark 3.2.8 Assume that p is an odd prime number with o2(p) odd. By Theorem 1.4.9, e2(Q(ζp)/Q) = 1 and f2(Q(ζp)/Q) = op(2) is odd. By Theo- rem 1.4.4, it follows that e2(F (ζp)/Q) and f2(F (ζp)/Q) being odd is equivalent with both the ramiﬁcation index e2(F (ζp)/Q(ζp)) and the residue degree f2(F (ζp)/Q(ζp)) being odd.

Following the proof of the previous proposition we have:

Proposition 3.2.9 Let M be a group in (Z) of odd order such that 2 has odd order modulo |M| and let F be an abelian number ﬁeld. Then Q8 × M is F -critical if and only if M is a cyclic group of prime order p, F is totally real and both e2(F (ζp)/Q) and f2(F (ζp)/Q) are odd. In this case F (Q8 × Cp) contains Ä ä an exceptional component −1,−1 . F (ζp)

Now we consider the Z-groups from Amitsur’s classiﬁcation. Amitsur sug- gests in his paper [Ami55] that one can discover a minimal faithful division algebra component in the Wedderburn decomposition of the groups in (Z).

We give some lemmas ﬁrst. Recall that we denote by Im(F ) the image of Gal(F (ζm)/F ) in U(Z/mZ). Because of this correspondence, we will often r abuse notation and denote an automorphism Q(ζm) → Q(ζm): ζm 7→ ζm by its deﬁning integer r. n/k Let G = haim ok hbin and A = ab . Then A is cyclic and normal and maximal abelian in G and hence (A, 1) is a strong Shoda pair of G. We n/k investigate the structure of the simple algebra F GeC (G, ab , 1), described n/k in Theorem 2.1.6, when EF (G, ab ) = G.

Lemma 3.2.10

Let F be a number ﬁeld and G = haim ok hbin with gcd(m, n) = 1, −1 r n n/k b ab = a and r an integer with om(r) = k . Then EF (G, ab ) = G hri if and only if Q(ζm) ∩ F is contained in Q(ζm) .

Proof Let r0 be such that r0 ≡ r mod m and r0 ≡ 1 mod k. Then, G = n/k 0 EF (G, ab ) if and only if hr i ⊆ Imk(F ). This happens if and only if r0 σ : ζmk 7→ ζmk is a Galois automorphism of the extension F (ζmk)/F , which is equivalent with F being in the fixed field of σ. Since σ fixes ζk, this again is hri equivalent with Q(ζm) ∩ F ⊆ Q(ζm) .

79 exceptional components

Lemma 3.2.11

Let F be a number ﬁeld and G = haim ok hbin with gcd(m, n) = 1, −1 r n hri b ab = a , r an integer such that om(r) = k and Q(ζm) ∩ F ⊆ Q(ζm) . n/k r Let A = ab , C ∈ CF (A), σ : F (ζmk) → F (ζmk): ζm 7→ ζm and hσi K = F (ζmk) . Then

F GeC (G, A, 1) = (K(ζm)/K, σ, ζk).

If furthermore n = 4 and k = 2, then r ≡ −1 mod m and

Å −1 2 ã −1, (ζm − ζm ) F GeC (G, A, 1) = −1 . F (ζm + ζm )

Proof Let A = abn/k . From Lemma 3.2.10 and Theorem 2.1.6 it follows that n −1 Xk F GeC (G, A, 1) = F (ζmk) ∗ G/A = F (ζmk)uσi i=0

ri n/k with uσi ζm = ζm uσi , uσi ζk = ζkuσi and uσi = ζk. The center of this simple hσi algebra is clearly equal to K = F (ζmk) and therefore we can denote it as the cyclic cyclotomic algebra F GeC (G, A, 1) = (K(ζm)/K, σ, ζk). If n = 4 and k = 2, then the degree of F GeC (G, A, 1) is 2 and hence it is a −1 quaternion algebra over its center F (ζm + ζm ). An easy computation shows that F GeC (G, A, 1) is generated over its center by elements x and y satisfying 2 2 −1 2 x = ζ2 = −1 and y = (ζm − ζm ) and xy = −yx. The following lemma comes from [CdR14, Lemma 2.3].

Lemma 3.2.12 (Caicedo-del R´ıo) −1 r Let G = haim ok hbin with gcd(m, n) = 1, b ab = a , r an integer such n n/k that om(r) = k and let A = ab . Then G is a subgroup of a division algebra D if and only if QGe(G, A, 1) is embedded in D.

Proof Let A = abn/k . By Lemma 3.2.11,

n −1 Xk QGe(G, A, 1) = Q(ζmk) ∗ G/A = Q(ζmk)uσi , i=0

80 3.2 group algebras with exceptional components of type ec1

n/k ri with uσi = ζk, uσi ζm = ζm uσi and uσi ζk = ζkuσi . Moreover, a 7→ ζm and b 7→ uσ determines an injective group homomorphism

G → U(QGe(G, A, 1)).

Let D be a division algebra such that f : G → U(D) is an injective group homomorphism. Then f(a) and f(bn/k) are commuting roots of unity of order −1 r m and k respectively and f(b ab) = f(a) . Thus ζm 7→ f(a) and uσ 7→ f(b) determines an algebra homomorphism Q(ζmk) ∗ G/A → D which is injective because Q(ζmk) ∗ G/A is simple. Proposition 3.2.13

Let F be a number ﬁeld and G = haim ok hbin with gcd(n, m) = 1. Let n/k A = ab and C ∈ CF (A). If F GeC (G, A, 1) is not a division algebra, then FG does not have a division algebra as a faithful simple component. Furthermore, if FG contains a division algebra D as a faithful simple component, then F GeC (G, A, 1) is embedded in D and in particular EF (G, A) = G, F (ζmk) ∗ G/A is a division algebra and D has degree at n least k .

n/k Proof Let A = ab , E = EF (G, A) and C ∈ CF (A). Then (A, 1) is a strong Shoda pair of G and, by Theorem 2.1.6,

F GeC (G, A, 1) = M[G:E](F (ζmk) ∗ E/A)

is a Wedderburn component of FG which is a direct factor of the semisimple

algebra F ⊗Q QGe(G, A, 1). It is easy to check that F GeC (G, A, 1) is a faithful component of FG. Assume now that FG contains as a faithful simple component a division algebra D not isomorphic to F GeC (G, A, 1). Then also QG contains a division algebra D0 not isomorphic to QGe(G, A, 1) as a faithful simple component. By Lemma 3.2.12, QGe(G, A, 1) is embedded in D0. But this means that F GeC (G, A, 1) is a direct factor of F ⊗Q QGe(G, A, 1), which in its turn is 0 embedded in F ⊗Q D . If F GeC (G, A, 1) is not a division algebra, then it is a 0 matrix ring and it has nilpotent elements. But then also F ⊗Q D has nilpotent 0 elements. Since F ⊗Q D is a direct sum of isomorphic copies of D, also D has nilpotent elements, which is a contradiction. n We conclude that F GeC (G, A, 1) is a division algebra of degree k , which is n embedded in D. Necessarily E = G and D has degree at least k .

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Lemma 3.2.14 If N ¢ G, (H, 1) is a strong Shoda pair of G and (HN/N, 1) is a strong Shoda pair of G/N, then EF (G, H)N/N ⊆ EF (G/N, HN/N).

i Proof This follows directly from the fact that if ζ|H| 7→ ζ|H| determines an automorphism of Gal(F (ζ|H|)/F ), then it restricts to an automorphism i ζ[HN:N] 7→ ζ[HN:N] in Gal(F (ζ[HN:N])/F ). Because of the structure of metacyclic groups and their group algebras, we can deduce that, for m a prime number, the above minimal faithful division algebra component is essentially the only possible faithful division algebra showing up in the Wedderburn decomposition. Proposition 3.2.15

Let F be a number ﬁeld, p a prime number and G = haip ok hbin with gcd(n, p) = 1. The only possible faithful division algebra components of n/k n/k FG are the algebras F GeC (G, ab , 1), with C ∈ CF ( ab ).

Proof Let b−1ab = ar. By Lemma 1.7.17, we deduce that the only non- commutative components of FG are determined by the strong Shoda pairs n d (Gd,K) with d a divisor of k diﬀerent from 1, Gd = a, b , Gd/K cyclic and n rx−1 n/k d = min{x | k : a ∈ K}. Let A = Gn/k = ab . Assume that a strong Shoda pair (Gd,K) leads to a faithful division algebra component of FG, then EF (G, Gd/K) = G and F GeC (G, Gd,K) = n F (ζ[Gd:K]) ∗ G/Gd has degree at least k by Proposition 3.2.13. However its n ln/k degree equals |G/Gd| = d. Therefore d = k , Gd = A and K = b ⊆ Z(G) for l | k. Since (A, K) is a Shoda pair and the characteristic subgroup K is contained in the kernel of the character associated to eC (G, A, K), in order for F GeC (G, A, K) to be a faithful component, K has to be 1. Clearly the groups of type (Z)(a) are never F -critical because the groups are abelian. We study the groups of type (Z)(b). For this, we need two lemmas. Lemma 3.2.16 Let F/Q be a ﬁnite normal extension such that F is totally real. Assume −1 2 ω = (ζm − ζm ) ∈ F for some positive integer m ≥ 3. Then ω is totally negative.

82 3.2 group algebras with exceptional components of type ec1

Proof Let σ : F → R be an embedding of F in R. Since F/Q is normal, 2 −2 σ(F ) = F . Also σ(ω) = −2 + σ(ζm + ζm ). Because F (ζm)/F is normal, the map σ extends to an automorphismσ ˆ : F (ζm) → F (ζm) in the Galois group r 2 −2 of F (ζm)/Q. Thereforeσ ˆ(ζm) = ζm 6= 1, for some integer r, and σ(ζm + ζm ) 2 −2 is again a sum of 2 roots of unity. Hence |σ(ζm + ζm )| < 2 and σ(ω) < 0. Lemma 3.2.17 Let F be a number ﬁeld and n, m two positive integers such that n divides −1 2 −1 2 −1,(ζn−ζn ) −1,(ζm−ζm ) m. If F splits −1 , then F also splits −1 . Q(ζn+ζn ) Q(ζm+ζm )

Proof Using the binomial formula and an induction argument, one easily k −k −1 −1 −1 proves that ζm + ζm ∈ Q(ζm + ζm ). Therefore, Q(ζn + ζn ) ⊆ Q(ζm + ζm ). Also, for any positive integer k, the following equality is well known:

xk − yk = (x − y)(xk−1 + xk−2y + ... + xyk−2 + yk−1).

k Applying this to ζn = ζm, we have

−1 −1 k−1 k−3 −k+3 −k+1 ζn − ζn = (ζm − ζm )(ζm + ζm + ... + ζm + ζm ). (11)

−1 2 −1,(ζn−ζn ) If F splits −1 , then by Proposition 1.2.1 there exists a triple Q(ζn+ζn ) −1 3 2 −1 2 2 2 (x, y, z) ∈ F (ζn + ζn ) \{(0, 0, 0)} such that −x + (ζn − ζn ) y = z . By 2 −1 2 2 2 (11), it follows that the equation −x − (ζm − ζm ) y = z holds for some −1 2 −1 3 −1,(ζm−ζm ) (x, y, z) ∈ F (ζm + ζm ) \{(0, 0, 0)} and hence F splits −1 . Q(ζm+ζm ) Theorem 3.2.18 Let F be an abelian number ﬁeld and let G be a ﬁnite group. Then G is a Z-group of type (Z)(b) and F -critical if and only if G = Cp o2 C4, with p a prime number satisfying p ≡ −1 mod 4, F is totally imaginary, −1 Q(ζp) ∩ F ⊆ Q(ζp + ζp ) and both ep(F/Q) and fp(F/Q) are odd. −1 2 −1,(ζp−ζp ) In this case FG contains an exceptional component −1 . F (ζp+ζp )

m 4 −1 −1 Proof Let G = Cm o2 C4 = a, b : a = 1, b = 1, b ab = a with m odd and let F be an abelian number ﬁeld. Assume that G is F -critical and let −1 2 2 −1,(ζm−ζm ) A = ab . Then EF (G, A) = G and F GeC (G, A, 1) = −1 is a F (ζm+ζm ) division algebra for any C ∈ CF (A) because of Lemma 3.2.11 and Proposi- tion 3.2.13. By Lemma 3.2.10, EF (G, A) = G is equivalent with Q(ζm) ∩ F ⊆ −1 Q(ζm + ζm ).

83 exceptional components

Assume that m is not prime and choose a prime divisor p of m. Then G/N = p haip o2 b 4 for N = ha i ¢ G. By Lemma 3.2.14, EF (G/N, A/N) = G/N. −1 2 −1,(ζp−ζp ) Therefore, for any D ∈ CF (A/N), F (G/N)eD(G/N, A/N, 1) = −1 F (ζp+ζp ) is itself a division algebra by Lemma 3.2.17. Since G is F -critical, the algebra F (G/N)eD(G/N, A/N, 1) has to be a totally deﬁnite quaternion algebra and −1 hence F (ζp + ζp ) is totally real. But this means that F is totally real. Since G is by assumption F -critical, one of the strong Shoda pairs (A, K) with K ⊆ A and a∈ / K should produce a division algebra which is not a totally deﬁnite quaternion algebra (see Lemma 1.7.17). The center of such −1 a F GeC (G, A, K) equals F (ζ[A:K] + ζ[A:K]) and is not totally real (since if −1 −1 2 F (ζ[A:K] + ζ[A:K]) is totally real then −1 and (ζ[A:K] − ζ[A:K]) are totally negative by Lemma 3.2.16). Therefore F is not totally real, a contradiction. Hence F is totally imaginary, m = p prime and

−1 −1 F GeC (G, A, 1) = (F (ζp)/F (ζp + ζp ), ζp 7→ ζp , −1) Ç −1 2 å −1, (ζp − ζp ) = −1 F (ζp + ζp ) Ç −1 2 å −1 −1, (ζp − ζp ) = F (ζp + ζ ) ⊗ −1 p Q(ζp+ζp ) −1 Q(ζp + ζp ) is a division algebra by Proposition 3.2.15. By Proposition 3.2.2 and Theo- −1 2 −1,(ζp−ζp ) rem 1.5.3, this means that mp −1 6= 1 and both the ramiﬁcation Q(ζp+ζp ) −1 −1 −1 −1 index ep(F (ζp + ζp )/Q(ζp + ζp )) and fp(F (ζp + ζp )/Q(ζp + ζp )) are odd. By Lemma 1.5.5 and Example 1.4.11:

Ç −1 2 å ® f ´ −1, (ζp − ζp ) p − 1 2 mp −1 = min l ∈ N : f ≡ 0 mod , Q(ζp + ζp ) gcd(p − 1, e) gcd(2, l)

−1 −1 where e = ep(Q(ζp)/Q(ζp + ζp )) = 2 and f = fp(Q(ζp + ζp )/Q) = 1. −1 2 −1,(ζp−ζp ) We see that mp −1 6= 1 if and only if p ≡ −1 mod 4. Also the Q(ζp+ζp ) −1 p−1 ramiﬁcation index ep(Q(ζp + ζp )/Q) = 2 is odd since p ≡ −1 mod 4, −1 −1 −1 fp(Q(ζp + ζp )/Q) = 1 and both ep(F (ζp + ζp )/F ) and fp(F (ζp + ζp )/F ) p−1 divide 2 which is odd. Therefore, by Theorem 1.4.4, the ramiﬁcation index −1 −1 −1 −1 ep(F (ζp+ζp )/Q(ζp+ζp )) and the residue degree fp(F (ζp+ζp )/Q(ζp+ζp )) are odd if and only if ep(F/Q) and fp(F/Q) are odd. We conclude that G is as in the statement of the theorem.

84 3.2 group algebras with exceptional components of type ec1

Assume now that G and F are as in the statement. Since p ≡ −1 mod 4, C4 acts by inversion on Cp and clearly, G is a Z-group of type (Z)(b). Let 2 −1 A = ab . Since Q(ζp) ∩ F ⊆ Q(ζp + ζp ), EF (G, A) = G by Lemma 3.2.10. −1 2 −1,(ζp−ζp ) Therefore F GeC (G, A, 1) = −1 for any C ∈ CF (A). This sim- F (ζp+ζp ) −1 ple component has degree 2 over its center F (ζp + ζp ), which is totally imaginary. Therefore it is not a totally definite quaternion algebra. Due −1 2 −1,(ζp−ζp ) to, p ≡ −1 mod 4, mp −1 6= 1 and because of the assumptions Q(ζp+ζp ) −1 −1 −1 −1 both ep(F (ζp + ζp )/Q(ζp + ζp )) and fp(F (ζp + ζp )/Q(ζp + ζp )) are odd. −1 2 −1,(ζp−ζp ) By Proposition 3.2.2, the simple component −1 is now a division F (ζp+ζp ) algebra. Furthermore G has only quotients isomorphic to cyclic and dihedral groups, which have only fields and matrix rings as simple components over F . Therefore G is F -critical. Lemma 3.2.19 Let G and H be finite groups with coprime order. If G × H is F -critical, then G or H is a cyclic group.

Proof Assume that G × H is F -critical, then the simple division algebra component equals F (G×H)e = F Ge1 ⊗F F He2 for some idempotents e, e1, e2 and both F Ge1 and F He2 are division algebras (including fields). If both F Ge1 and F He2 are non-commutative division algebras, then at least one of F Ge1 and F He2 is of odd degree because G and H have coprime order. Therefore, they can not be both a totally definite quaternion algebra. This means that G × H can never be F -critical, unless either G = 1 or H = 1. If one of both, say F Ge1, is a commutative division ring and thus a field, 0 then F Ge1 is a Wedderburn component of F (G/G ). Because of the F -critical condition, we can assume that G0 = 1 and hence G is an abelian group in (Z). Since abelian groups in (Z) are cyclic, G is cyclic. Theorem 3.2.20 Let F be an abelian number field and G a group. Then G is a Z-group of type (Z)(c) and F -critical if and only if one of the following holds:

a) G = Cq ×(Cp o2 C4), with q and p diﬀerent odd prime numbers with oq(p) odd and p ≡ −1 mod 4. Moreover F is totally real and both ep(F (ζq)/Q) and fp(F (ζq)/Q) are odd. In this case FG contains an −1 2 −1,(ζp−ζp ) exceptional component −1 ; F (ζq ,ζp+ζp )

85 exceptional components

b) G = haip ok hbin, with n ≥ 8, p an odd prime number not dividing n, −1 r n b ab = a , and both k and k are divisible by all the prime numbers r rn/k−1 n dividing n. Moreover Q(ζp)∩F ⊆ Q(ζp +ζp +...+ζp ), mp = k and one of the following holds:

i. either p ≡ 1 mod 4 or n is odd, vq(p − 1) ≤ vq(k) for every n prime divisor q of n and mp,h < k for every h 6= k divisor of k such that vq(p − 1) ≤ vq(h) for every prime divisor q of n. If n = 2k and F is totally imaginary, then mp,2 = 1.

ii. p ≡ −1 mod 4, v2(k) = 1, v2(n) = 2, vq(p − 1) ≤ vq(k) for n every odd prime divisor q of n and mp,h < k for every h 6= k divisor of k such that v2(h) = 1 and vq(p − 1) ≤ vq(h) for every odd prime divisor q of n.

iii. p ≡ −1 mod 4, v2(p + 1) + 1 ≤ v2(k), v2(n) = v2(k) + 1, vq(p − 1) ≤ vq(k) for every odd prime divisor q of n and n (1) mp,h < k for every divisor h of k diﬀerent from k such that v2(p + 1) + 1 ≤ v2(h) and vq(p − 1) ≤ vq(k) for every odd prime divisor q of n, n (2) mp,h < k for every divisor h of k diﬀerent from 2 and k such that v2(h) = 1 and vq(p − 1) ≤ vq(h) for every odd prime divisor q of n. If n = 2k and F is totally imaginary, then mp,2 = 1. Here ® pf − 1 k ´ m = min l ∈ : ≡ 0 mod p N gcd(pf − 1, e) gcd(k, l)

r rn/k−1 with K = F (ζk, ζp + ζp + ... + ζp ), e = ep(F (ζpk)/K) and f = fp(K/Q), and ® ´ pfh − 1 h mp,h = min l ∈ N : f ≡ 0 mod gcd(p h − 1, eh) gcd(h, l)

with Kh = K ∩ F (ζph), eh = ep(F (ζph)/Kh) and fh = fp(Kh/Q).

In this case, FG contains (K(ζp)/K, σ, ζk) as an exceptional component, where the action is deﬁned by r σ : F (ζpk) → F (ζpk): ζp 7→ ζp ; ζk 7→ ζk.

86 3.2 group algebras with exceptional components of type ec1

Proof Assume that G is a Z-group of type (Z)(c) which is F -critical. Then

Y G = C C = C R = C ×(P R )×· · ·×(P R ), m0 ok n m0 ok p m p1 okp1 p1 ps okps ps p|m0

with k = kp1 ··· kps and p1, . . . , ps the prime divisors of m0 such that Cn does

not act trivially on Ppi . Here all pi are odd since p = 2 can never satisfy (3) of Remark 1.10.2. By Lemma 3.2.19, necessarily G = Cm × (haipt ok hbin) with p odd and gcd(m, n) = 1 = gcd(m, p) = gcd(n, p). n/k We claim that t = 1. Assume that t 6= 1 and let A = Cm × ab . Then EF (G, A) = G and F GeC (G, A, 1) = F (ζmptk) ∗ G/A is a division algebra for any C ∈ CF (A) because of Proposition 3.2.13 and Lemma 3.2.11. p Take N = ha i. Then G/N = Cm × (haip ok b n), EF (G/N, A/N) = G/N, by Lemma 3.2.14, and, for any D ∈ CF (A/N), F (G/N)eD(G/N, A/N, 1) = F (ζmpk) ∗ G/A is naturally embedded in F GeC (G, A, 1). Hence also F (G/N) n contains a simple component which is a division algebra of degree k . Therefore n it should be a totally deﬁnite quaternion algebra and hence k = 2. This means −1 −1 that the action of b on hai is of order 2 and b ab = a . Since F (ζmk) is contained in the center of F (ζmpk) ∗ G/A, which is totally real, m = 1 and k ≤ 2. By Remark 1.10.2, k 6= 1. So k = 2 and n = 4. Furthermore, F is totally real. Now G is as in case (Z)(b), but by Theorem 3.2.18, F cannot be totally real in order for G to be F -critical. This contradiction tells us that t = 1.

From now on G = Cm × (haip ok hbin) is F -critical and we distinguish two cases, m 6= 1 and m = 1.

Assume ﬁrst that m 6= 1 and G = Cm × (haip ok hbin) is F -critical. We n/k claim that m is a prime number. Let N = Cm and A = Cm × ab .

Then G/N = haip ok b n, EF (G/N, A/N) = G/N by Lemma 3.2.14 and, for any D ∈ CF (A/N), F (G/N)eD(G/N, A/N, 1) = F (ζpk) ∗ G/A. It is naturally embedded in F (ζmpk) ∗ G/A and therefore it is again a division alge- n bra of degree k . Therefore it should be a totally deﬁnite quaternion algebra n and hence k = 2. As before, we conclude that k = 2, n = 4 and F is totally real. Now for some normal subgroup M of G and some prime divi-

sor q of m, G/M = Cq × (haip o2 b 4). Then F (G/M)eD(G/M, A/M, 1) = −1 2 −1,(ζp−ζp ) F (ζ2qp) ∗ G/A = −1 should be a totally deﬁnite quaternion alge- F (ζq ,ζp+ζp ) bra, but its center contains F (ζq) which is not totally real. This is a contradiction and hence m is a prime number.

87 exceptional components

So we can assume that G = Cq × (haip o2 hbi4) with q and p diﬀerent odd prime numbers. Using the conditions of (Z)(c), we get 2 = 2o2(p) - oq(p). Hence oq(p) is odd. Moreover, if p ≡ 1 mod 4, then 1 = v2(k) ≥ v2(p−1) ≥ 2, a contradiction. Thus p ≡ −1 mod 4. Now

−1 −1 F GeC (G, A, 1) = (F (ζqp)/F (ζq, ζp + ζp ), ζp 7→ ζp , −1) Ç −1 2 å −1, (ζp − ζp ) = −1 F (ζq, ζp + ζp ) Ç −1 2 å −1 −1, (ζp − ζp ) = F (ζq, ζp + ζ ) ⊗ −1 p Q(ζp+ζp ) −1 Q(ζp + ζp )

is a division algebra. By Theorem 1.5.3 and Proposition 3.2.2, the local in- −1 2 −1,(ζp−ζp ) −1 −1 dex mp −1 6= 1 and both ep(F (ζq, ζp + ζp )/Q(ζp + ζp )) and Q(ζp+ζp ) −1 −1 fp(F (ζq, ζp + ζp )/Q(ζp + ζp )) are odd. −1 −1 By Example 1.4.11, ep(Q(ζp)/Q(ζp + ζp )) = 2 and fp(Q(ζp + ζp )/Q) = 1, −1 2 −1,(ζp−ζp ) hence it follows from Lemma 1.5.5 that mp −1 6= 1 if and only Q(ζp+ζp ) −1 if p ≡ −1 mod 4. Since p ≡ −1 mod 4, [Q(ζp + ζp ): Q] is odd and −1 −1 −1 also ep(Q(ζp + ζp )/Q), fp(Q(ζp + ζp )/Q), ep(F (ζq, ζp + ζp )/F (ζq)) and −1 −1 −1 fp(F (ζq, ζp + ζp )/F (ζq)) are odd. Therefore ep(F (ζq, ζp + ζp )/Q(ζp + ζp )) −1 −1 and fp(F (ζq, ζp + ζp )/Q(ζp + ζp )) are odd if and only if ep(F (ζq)/Q) and fp(F (ζq)/Q) are odd. This means that G and F are as in (a).

Conversely, assume that G and F are as in (a), then G = Cpq o2 C4 = 2 Cq × (haip o2 hbi4) is a Z-group of type (c). Let A = Cq × ab and −1 C ∈ CF (A). Since F is totally real, clearly Q(ζp) ∩ F ⊆ Q(ζp + ζp ) and G = EF (G, A) by Lemma 3.2.10. Since F GeC (G, A, 1) has a totally imag- −1 inary center F (ζq, ζp + ζp ), it is not a totally definite quaternion algebra. −1 2 −1,(ζp−ζp ) Due to p ≡ −1 mod 4, mp −1 6= 1 and because of the assumptions Q(ζp+ζp ) −1 −1 −1 −1 both ep(F (ζq, ζp + ζp )/Q(ζp + ζp )) and fp(F (ζq, ζp + ζp )/Q(ζp + ζp )) are odd. Because of Proposition 3.2.2, F GeC (G, A, 1) is now a division algebra. Therefore FG contains an exceptional component of type EC1. The proper non-abelian quotients of G are Cp o2 C4,D2p,D2pq and, since F is totally real, those groups give rise to simple components which are either fields, matrix rings or totally definite quaternion algebras. Therefore G is F -critical.

88 3.2 group algebras with exceptional components of type ec1

Assume now that m = 1 and G = a, b : ap = 1, bn = 1, b−1ab = ar is F - n critical, with p an odd prime number, gcd(p, n) = 1 and op(r) = k . By Remark 1.10.2, if q is a prime divisor of n then n 1 ≤ v ( ) ≤ v (p − 1) ≤ v (k). q k q q

In particular vq(n) ≥ 2, so either n = 4 and G = Cp o2 C4 or n ≥ 8. How- ever G = Cp o2 C4 is of type (Z)(b), thus we can assume that n ≥ 8. Let A = abn/k . By assumption, FG contains an exceptional component of type EC1, so by Proposition 3.2.13, F GeC (G, A, 1) is a division algebra which has n degree k , and in particular EF (G, A) = G. By Lemmas 3.2.10 and 1.4.13, hri r rn/k−1 EF (G, A) = G is equivalent to Q(ζp)∩F ⊆ Q(ζp) = Q(ζp +ζp +...+ζp ). From the conditions on (Z)(c), if n is even, we have 2 ≤ v2(p − 1) ≤ v2(k) when p ≡ 1 mod 4 and either v2(k) = 1 or 3 ≤ v2(p + 1) + 1 ≤ v2(k) when n p ≡ −1 mod 4. Due to k divides p−1 (see Remark 1.10.2), if p ≡ −1 mod 4, then v2(p − 1) = 1 and v2(n) = v2(k) + 1. Together with the other conditions from (Z)(c) on the parameters of G, this gives rise to the following cases:

i. either p ≡ 1 mod 4 or n is odd, vq(p−1) ≤ vq(k) for every prime divisor q of n;

ii. p ≡ −1 mod 4, v2(k) = 1, v2(n) = 2 and vq(p − 1) ≤ vq(k) for every odd prime divisor q of n;

iii. p ≡ −1 mod 4, 3 ≤ v2(p + 1) + 1 ≤ v2(k), v2(n) = v2(k) + 1 and vq(p − 1) ≤ vq(k) for every odd prime divisor q of n.

r Let σ be the automorphism of F (ζpk) which maps ζp to ζp and ﬁxes F (ζk). hσi Let K = F (ζpk) . By our assumptions and Lemma 3.2.11, for any C ∈ CF (A) we have F GeC (G, A, 1) = (K(ζp)/K, σ, ζk). Since F GeC (G, A, 1) is a division algebra, n = ind(F Ge (G, A, 1)) = lcm(m (F Ge (G, A, 1)) : q = p, ∞), k C q C

by Theorems 1.4.3 and 1.5.3. If m∞(F GeC (G, A, 1)) = 2, then K ⊆ R by Lemma 1.5.4. Note that always ζk ∈ K, so k = 2. This implies that n is a power of 2. However, when k = 2, G is in case (ii) and v2(n) = 2, a contradiction because of n ≥ 8. Thus, m∞(F GeC (G, A, 1)) = 1. By Lemma 1.5.5, mp(F GeC (G, A, 1)) = mp as in the statement of the theorem. Therefore

89 exceptional components

n k = ind(F GeC (G, A, 1)) = mp and F GeC (G, A, 1) is an exceptional component of FG (see Lemma 3.2.1). For each of the cases (i)-(iii), let h | k, h 6= k be an integer satisfying the conditions as in the statement of the theorem or h = 2 when k is even, and ¨ hn ∂ let N = b k ⊆ Z(G). Then G/N = Cp h C hn is a non-abelian proper o k hn quotient of G. Note also that the prime divisors of n and k are the same n since the prime divisors of n and k are the same. Since N ⊆ Z(G), the images of the actions of Cn on Cp and of C hn on Cp are the same. So each k Sylow q-subgroup of C hn acts non-trivial on Cp and G/N is again a Z-group k of type (Z)(b) or (Z)(c). By our assumption on G, G/N cannot have exceptional components. Therefore the simple component F (G/N)eD(G/N, A/N, 1) n of degree k of F (G/N) is not exceptional for any D ∈ CF (A/N). Since EF (G, A) = G, by Lemma 3.2.14, EF (G/N, A/N) = G/N, and then by Lemma 3.2.11, F (G/N)eD(G/N, A/N, 1) = (Kh(ζp)/Kh, σ, ζh) with ζh ∈ Kh. We also use σ to denote its restriction to F (ζph). Hence

ind(F (G/N)eD(G/N, A/N, 1)) n = lcm(m (F (G/N)e (G/N, A/N, 1)) : q = p, ∞) ≤ q D k by Theorems 1.4.3 and 1.5.3. As we assume G is F -critical, the algebra F (G/N)eD(G/N, A/N, 1) is either a matrix ring or a totally deﬁnite quaternion algebra. If F (G/N)eD(G/N, A/N, 1) is a totally deﬁnite quaternion al- n gebra, then the degree k = 2 and Kh ⊆ R, so h = 2 and G/N = Cp o2 C4. In this case F is totally real. If F (G/N)eD(G/N, A/N, 1) is a matrix ring, n then lcm(mq(F (G/N)eD(G/N, A/N, 1)) : q = p, ∞) < k . We claim that m∞(F (G/N)eD(G/N, A/N, 1) = 1. Suppose that this local index equals 2, hn n then Kh ⊆ R by Lemma 1.5.4, thus h = 2. It follows that k = 4 and k = 2. So

lcm(mq(F (G/N)eD(G/N, A/N, 1)) : q = p, ∞) = 1,

and hence the local index m∞(F (G/N)eD(G/N, A/N, 1)) = 1, a contradiction. Thus the local index m∞(F (G/N)eD(G/N, A/N, 1)) = 1 and hence n mp(F (G/N)eD(G/N, A/N, 1) = mp,h < k , with mp,h the formula as in the statement of the theorem because of Janusz’ formula in Lemma 1.5.5.

Suppose that h 6= 2, then Kh is not totally real and hence the algebra n F (G/N)eD(G/N, A/N, 1) is a matrix ring and mp,h < k . If h = 2, then n = 2k and G/N = Cp o2 C4. If F (G/N)eD(G/N, A/N, 1) is a matrix ring,

90 3.2 group algebras with exceptional components of type ec1

then mp,2 < 2, so mp,2 = 1. If F (G/N)eD(G/N, A/N, 1) is a totally deﬁnite quaternion algebra, then F is totally real. Hence G is as in (b). Conversely, assume now that G and F are as in (b). Since any prime divisor n q of n divides k , the order of the image of the action of Cn on Cp, any Sy- low q-subgroup of Cn acts non-trivial on Cp. Together with the assumptions on n and k in (i)-(iii), this means that G is a Z-group of type (Z)(c). Let n/k A = ab . By Lemma 3.2.10 and the assumptions on F , EF (G, A) = G. Then by Lemma 3.2.11, F GeC (G, A, 1) = (K(ζp)/K, σ, ζk) for any C ∈ CF (A). n Furthermore, F GeC (G, A, 1) has degree k and always ζk ∈ K. By Theo- rems 1.4.3 and 1.5.3,

ind(F GeC (G, A, 1)) = lcm(mq(F GeC (G, A, 1)) : q = p, ∞).

We claim that m∞(F GeC (G, A, 1)) = 1. Assume m∞(F GeC (G, A, 1)) = 2, then K ⊆ R by Lemma 1.5.4, and hence k = 2 and n is a power of 2. So G is as in (i) or (iii), but from both conditions we can deduce that k cannot be 2. Thus, m∞(F GeC (G, A, 1)) = 1. Therefore ind(F GeC (G, A, 1)) = mp(F GeC (G, A, 1)) and by Lemma 1.5.5 and the assumptions, it follows that n mp(F GeC (G, A, 1)) = mp = k . By Lemma 3.2.1, F GeC (G, A, 1) is an exceptional component of type EC1. In order to prove that G does not have proper quotients with exceptional components of type EC1 in their Wedderburn decomposition over F , we argue by means of contradiction. Let N be a normal subgroup of G such that F (G/N) contains an exceptional component of type EC1. Then G/N is non- ¨ hn ∂ abelian and hence N = b k ⊆ Z(G) for some divisor h of k, h 6= k. Then

G/N = Cp h C hn and without loss of generality we can assume that this o k group contains a faithful exceptional component of type EC1. Then G/N is as in (Z)(b) or (Z)(c), and by Proposition 3.2.15, F (G/N)eD(G/N, A/N, 1) = (Kh(ζp)/Kh, σ, ζh) is an exceptional division algebra. Hence n = ind(F (G/N)e (G/N, A/N, 1)) = k D lcm(mq(F (G/N)eD(G/N, A/N, 1)) : q = p, ∞).

If G/N is as in (Z)(b), then G/N = Cp o2 C4, h = 2 and n = 2k. So G is as in (i) or (iii) and by the assumptions on G, F is totally real or mp,2 = 1. By Lemma 3.2.11, Ç −1 2 å −1, (ζp − ζp ) F (G/N)eD(G/N, A/N, 1) = −1 . F (ζp + ζp )

91 exceptional components

We assume that it is exceptional, so

lcm(mq(F (G/N)eD(G/N, A/N, 1)) : q = p, ∞) = 2.

−1 2 −1,(ζp−ζp ) If F is totally real, then −1 is a totally definite quaternion algebra, F (ζp+ζp ) a contradiction. Hence mp(F (G/N)eD(G/N, A/N, 1)) = mp,2 = 1, but then m∞(F (G/N)eD(G/N, A/N, 1)) = 2 and F (G/N)eD(G/N, A/N, 1) is not an exceptional component by Lemma 3.2.1, again a contradiction. Hence G/N is as in (Z)(c). If m∞(F (G/N)eD(G/N, A/N, 1)) = 2, then Kh ⊆ R by Lemma 1.5.4, and then h = 2. However, h = 2 implies that n hn hn is a power of 2, p ≡ −1 mod 4 and v2( k ) = 2. It follows that k = 4 and G/N = Cp o2 C4, a contradiction. So, m∞(F (G/N)eD(G/N, A/N, 1)) = 1. hn Suppose that p ≡ 1 mod 4 or k is odd (equivalently n is odd; case (i)), hn then vq(p − 1) ≤ vq(h) for all prime divisors q of k (which are exactly the prime divisors of n). Assume first that h 6= 2, then by the assumptions, n mp(F (G/N)eD(G/N, A/N, 1)) = mp,h < k , a contradiction. Now regard the case when h = 2, then necessarily n is a power of 2 and G/N = Cp o2 C4, a contradiction. hn Assume that p ≡ −1 mod 4 and k is even (equivalently n is even). Then vq(p − 1) ≤ vq(h) for all odd prime divisors q of n. Also, either v2(k) = 1 or v2(p + 1) + 1 ≤ v2(k). We first deal with v2(k) = 1 (case (ii)). In this case, v2(n) = 2, v2(h) = 1 and h 6= 2 (as otherwise n = 4). So n mp(F (G/N)eD(G/N, A/N, 1)) < k , a contradiction. Finally, suppose that v2(p + 1) + 1 ≤ v2(k) (case (iii)). Then either v2(p + 1) + 1 ≤ v2(h) or n v2(h) = 1. If h 6= 2, then mp(F (G/N)eD(G/N, A/N, 1)) < k , a contradiction. When h = 2, G/N = Cp o2 C4 gives again a contradiction. We conclude that G does not have proper quotients with exceptional components of type EC1, hence G is F -critical. We summarize our results on F -critical groups. Theorem 3.2.21 Let D be a division ring and F an abelian number field, p and q different odd prime numbers. Then D is a Wedderburn component of FG for an F -critical group G if and only if one of the following holds:

−1,−1 (a) D = F , G ∈ {SL(2, 3),Q8}, F is totally imaginary and both, e2(F/Q) and f2(F/Q), are odd;

92 3.3 examples

Ä −1,−1 ä (b) D = , G ∈ {SL(2, 3)×Cp,Q8 ×Cp}, gcd(p, |G|/p) = 1, op(2) F (ζp) is odd, F is totally real and both, e2(F (ζp)/Q) and f2(F (ζp)/Q), are odd;

−1 2 −1,(ζp−ζp ) (c) D = −1 , G = Cp o2 C4, p ≡ −1 mod 4, F totally F (ζp+ζp ) −1 imaginary, Q(ζp)∩F ⊆ Q(ζp +ζp ) and both, ep(F/Q) and fp(F/Q), are odd;

−1 2 −1,(ζp−ζp ) (d) D = −1 , G = Cq × (Cp o2 C4), p ≡ −1 mod 4, oq(p) F (ζq ,ζp+ζp ) odd, F is totally real and both, ep(F (ζq)/Q) and fp(F (ζq)/Q), are odd;

® f ´ p − 1 k n min l ∈ N ≡ 0 mod = gcd(pf − 1, e) gcd(k, l) k

with e = ep(F (ζpk)/K) and f = fp(K/Q).

Proof This follows by combining the results from Theorems 1.10.1, 3.2.18 and 3.2.20 and Propositions 3.2.4 to 3.2.7 and 3.2.9.

3.3 examples

We apply our results to the case when F is Q. This is a restatement of the main result from [CdR14]. Corollary 3.3.1 (Caicedo-del R´ıo) Let G be a ﬁnite group.

1. QG contains an exceptional component of type EC1 if and only if G contains an epimorphic image H and one of the following holds:

93 exceptional components

a) H = Cq ×(Cp o2 C4), with p and q diﬀerent odd prime numbers, oq(p) is odd and p ≡ −1 mod 4;

b) H = haip ok hbin, with n ≥ 8, p an odd prime number not n dividing n, both k and k are divisible by all the prime numbers dividing n and one of the following holds: i. k = gcd(n, p − 1) and either p ≡ 1 mod 4 or n is odd;

ii. k = gcd(n, p − 1), p ≡ −1 mod 4 and v2(n) = 2; iii. p ≡ −1 mod 4, n = 2v2(p+1)+2 and k = 2v2(p+1)+1;

c) H = Q8 × Cp, with p an odd prime number and op(2) odd;

d) H = SL(2, 3)×Cp, with p a prime number diﬀerent from 2 and 3 and op(2) odd;

2. QG contains an exceptional component B of type EC2 if and only if G has an epimorphic image H such that the line H, Q, B appears in Table 2.

Moreover, QG has exactly the following exceptional components of type EC1:

−1 2 −1,(ζp−ζp ) −1 if G has an epimorphic image in case (a); Q(ζq ,ζp+ζp ) r rn/k−1 −1 (Q(ζkp)/Q(ζk, ζp+ζp +...+ζp ), σ, ζk), with r such that b ab = r r a and σ : ζp 7→ ζp , if G has an epimorphic image in case (b); Ä ä −1,−1 if G has an epimorphic image in case (c) or (d). Q(ζp)

Proof This is readily veriﬁed using Theorems 3.1.2 and 3.2.21, except for the fact why (i), (ii), (iii) is equivalent with (i), (ii), (iii) from Theorem 3.2.20. We use the notations from Theorem 3.2.20. First note that f = fp(K/Q) = n r rn/k−1 ok(p) and e = ep(Q(ζpk)/K) = op(r) = k for K = Q(ζk, ζp +ζp +...+ζp ) by Theorem 1.4.9. Assume that either p ≡ 1 mod 4 or n is odd. We prove that k = gcd(n, p−1) n if and only if mp = k , vq(p − 1) ≤ vq(k) for every prime divisor q of n and n mp,h < k for every h 6= k divisor of k such that vq(p − 1) ≤ vq(h) for every n p−1 prime divisor q of n. Suppose that k = gcd(n, p−1), then gcd( k , k ) = 1 and n p−1 since each prime divisor of n also divides k , also gcd(n, k ) = 1. Therefore,

94 3.3 examples

vq(p − 1) = vq(k) for every prime divisor q of n. Also ok(p) = 1 and hence k p−1 k n mp = min{l ∈ N : | n } = . Since vq(p − 1) = vq(k), for gcd(k,l) k gcd( k ,p−1) k each prime divisor q of n, there does not exist any proper divisor h of k such that vq(p − 1) ≤ vq(h), so we do not have to prove anything for those divisors. Conversely, since vq(p − 1) ≤ vq(k) for every prime divisor q of n, it already follows that gcd(n, p − 1) | k. Assume now that k 6= gcd(n, p − 1). Then h = gcd(n, p − 1) is a proper divisor of k with vq(p − 1) = vq(h) for every n prime divisor q of n. By the above, also mp,h = k , a contradiction. Assume now that p ≡ −1 mod 4 and v2(n) = 2. Using the same arguments n as in the previous case, we prove that k = gcd(n, p − 1) if and only if mp = k , n vq(p − 1) ≤ vq(k) for every odd prime divisor q of n and mp,h < k for every h 6= k divisor of k such that v2(h) = 1 and vq(p − 1) ≤ vq(h) for every odd prime divisor q of n. At last, assume that p ≡ −1 mod 4. We prove that n = 2v2(p+1)+2 and v2(p+1)+1 n k = 2 if and only if mp = k , v2(p + 1) + 1 ≤ v2(k), v2(n) = v2(k) + 1, vq(p − 1) ≤ vq(k) for every odd prime divisor q of n and

n (1) mp,h < k for every divisor h of k diﬀerent from k such that v2(p+1)+1 ≤ v2(h) and vq(p − 1) ≤ vq(k) for every odd prime divisor q of n,

n (2) mp,h < k for every divisor h of k diﬀerent from 2 and k such that v2(h) = 1 and vq(p − 1) ≤ vq(h) for every odd prime divisor q of n.

v2(p+1)+2 v2(p+1)+1 Assume that n = 2 and k = 2 , then clearly v2(p+1)+1 = v2(k), v2(n) = v2(k) + 1, vq(p − 1) ≤ vq(k) for every odd (none!) prime divisor q of n. Because of the structure of U(Z/kZ), we know that ok(p) = v2(p + 1) ok(p) ok(p) k p −1 n and 2k - p − 1. Therefore mp = min{l ∈ N : gcd(k,l) | 2 } = 2 = k . Since v2(p + 1) + 1 = v2(k), there does not exist any proper divisor h of k such that v2(p + 1) + 1 ≤ v2(h), so we do not have to prove anything for those divisors. Also, if v2(h) = 1, then h = 2, so we are ﬁnished. Conversely, we claim that v2(k) = v2(p + 1) + 1 and vq(k) = vq(p − 1) for every odd prime divisor q of n. Assume ﬁrst that v2(k) > v2(p + 1) + 1, then there exists a proper divisor h of k with v2(h) = v2(p + 1) + 1 and n vq(h) = vq(p − 1). One computes that mp,h = k , a contradiction by (1). Assume now that vq(k) > vq(p − 1) for some prime divisor q of n. Then there exists a proper prime divisor h of k with v2(h) = 1 and vq(p − 1) = vq(h). n Hence h = gcd(p − 1, n) and by the above mp,h = k . By (2), it follows that n h = 2. But then k = mp,h ≤ h = 2 and k is a power of 2, a contradiction.

95 exceptional components

So, v2(k) = v2(p + 1) + 1 and vq(k) = vq(p − 1) for every odd prime divisor v2(p+1)+2 v2(p+1)+1 q of n. Set n = 2 n1 with 2 - n1. Then k = 2 gcd(n1, p − 1). Suppose that gcd(n1, p − 1) 6= 1, then h = 2 gcd(n1, p − 1) = gcd(n, p − 1) is a proper divisor of k different from 2. Moreover, v2(h) = 1 and vq(p−1) ≤ vq(h) n for every odd prime divisor q of n. By the above, mp,h = k , which contradicts n n (2). Hence gcd(n1, p − 1) = 1. Since k divides p − 1 and both k and k are divisible by all the prime numbers dividing n, necessarily n = 1 and the result follows. The conditions of Theorem 3.2.21 are easy to check algorithmically and we did implement it in GAP. With this program we can compute the F -critical groups for any abelian number field F up to a fixed order. As an illustration we include the F -critical groups up to order 200 for all subfields of Q(ζ7). We compute the Schur index of the corresponding exceptional component A and we denote the center of A in the standard GAP notation. A local Schur index [p, s] means that mp(A) = s and mq(A) = 1 for all other prime numbers q. When for a fixed group, there are multiple lines in the table, this means that the exceptional component A appears as several isomorphic copies in the Wedderburn decomposition.

Table 3: List of Q(ζ7)-critical groups of type EC1 up to order 200 ID Structure Center Schur index Local index

[8, 4] Q8 CF(7) 2 [2, 2] [24, 3] SL(2, 3) CF(7) 2 [2, 2] [44, 1] C11 o2 C4 NF(77,[ 1, 43 ]) 2 [11, 2] [48, 1] C3 o8 C16 CF(56) 2 [3, 2] [80, 1] C5 o8 C16 NF(280,[ 1, 169 ]) 2 [5, 2] [92, 1] C23 o2 C4 NF(161,[ 1, 22 ]) 2 [23, 2] [117, 1] C13 o3 C9 NF(273,[ 1, 22, 211 ]) 3 [13, 3] [160, 3] C5 o8 C32 CF(56) 4 [5, 4] [172, 1] C43 o2 C4 NF(301,[ 1, 85 ]) 2 [43, 2]

96 3.3 examples

Table 4: List of Q-critical groups of type EC1 up to order 200 ID Structure Center Schur index Local index

[40, 1] C5 o4 C8 NF(20,[ 1, 9 ]) 2 [5, 2] [48, 1] C3 o8 C16 CF(8) 2 [3, 2] [56, 10] C7 × Q8 CF(7) 2 [2, 2] [63, 1] C7 o3 C9 NF(21,[ 1, 4, 16 ]) 3 [7, 3] [80, 3] C5 o4 C16 GaussianRationals 4 [5, 4] [84, 4] C3 × (C7 o2 C4) NF(21,[ 1, 13 ]) 2 [7, 2] [104, 1] C13 o4 C8 NF(52,[ 1, 25 ]) 2 [13, 2] [117, 1] C13 o3 C9 NF(39,[ 1, 16, 22 ]) 3 [13, 3] [132, 1] C11 × (C3 o2 C4) CF(11) 2 [3, 2] [156, 3] C13 × (C3 o2 C4) CF(13) 2 [3, 2] [168, 22] C7 × SL(2, 3) CF(7) 2 [2, 2] [176, 1] C11 o8 C16 NF(88,[ 1, 65 ]) 2 [11, 2] [184, 10] C23 × Q8 CF(23) 2 [2, 2]

97 exceptional components 14 10] [184, 1] [176, 22] [168, 16 3] [156, 1] [132, 1] [117, 1] [104, DSrcueCne cu ne oa index Local index Schur 4] [84, Center 3] [80, 10] [56, Structure 1] [48, 1] [40, ID C C C C C C C C C C C C Ls of List 5: Table 23 11 7 13 11 13 13 3 5 7 3 5 × × o × o o × o × × o o 4 8 4 SL(2 ( Q 8 3 4 C Q ( ( C C C C C 8 C C C 7 16 16 8 8 3 3 16 9 8 o , o o )C()2[,2] [2, 2 CF(7) 3) 2 2 2 C C C 4 4 4 F2, ,1 )2[,2] [7, 2 ]) 13 1, NF(21,[ ) Q F9, ,2 )2[,2] [3, 2 ]) 27 1, NF(91,[ ) F7, ,3 )2[,2] [3, 2 ]) 34 1, NF(77,[ ) ( ζ 7 F11[1 3 )2[,2] [2, 2 2] [11, 2 ]) 139 ]) 1, 505 NF(161,[ 265, 153, 1, NF(616,[ F7 2 2] [2, 2] [2, 2 2 CF(7) CF(7) F7 2 2] [2, 2] 2] [2, [13, 4] [5, 2] 3] [2, [13, 2 2 2 4 2 3 2] [3, ]) 211 2 139, 2] 118, [5, 55, 22, ]) 1, 337 NF(273,[ 209, 181, 1, NF(364,[ 2 ]) 13 1, NF(28,[ CF(7) CF(7) CF(7) ]) 41 ]) 1, 69 NF(56,[ 41, 29, 1, NF(140,[ F2, ,1 )2[,2] [7, 2] [7, 2 2 ]) 13 1, NF(21,[ ]) 13 1, NF(21,[ + ζ 7 − 1 -rtclgop ftp C pt re 200 order to up EC1 type of groups )-critical

98 3.3 examples 7)-critical groups of type EC1 up to order 200 − √ ( Q NF(7,[ 1, 2, 4 ])NF(77,[ 1, 23, 32,NF(56,[ 43, 1, 65, 9, 67 25 ])NF(21,[ ]) 1, 4, 16NF(21,[ ]) 1, 4, 16NF(280,[ ]) 1, 9, 81,NF(161,[ 121, 1, 169, 22, 249 93, ])NF(273,[ 114, 1, 116, 16, 137 22, ])NF(56,[ 79, 1, 100, 9, 172, 25 211,NF(301,[ ]) 235, 1, 256 44, ]) 85, 128, 130, 214 ]) 2 3 2 2 [11, 2] 2 [13, 3] 2 [2, 2] 3 [5, 2] [23, 2 3 2] [3, 2] [7, 3] [43, [7, 2] 3] 4 [5, 4] 4 4 9 4 16 9 16 32 C C C C C C C C 2 2 3 2 3) NF(7,[ 1, 2, 4 ]) 2 [2, 2] Table 6: List of 8 3 8 8 , o o o o o o o o 8 11 3 7 5 23 13 5 43 Q C C C C C C C C ID[8, 4] [24, Structure 3][44, 1] Center SL(2 [48, 1] [63, 1] [80, 1] [92, 1] [117, 1] [160, 3] [172, 1] Schur index Local index

CENTRALUNITS

In this chapter, we study Z(U(ZG)), the group of central units of ZG for finite groups G. Let Γ be a finitely generated abelian group. Assume that Γ has rank r and let u1, . . . , ur ∈ Γ. Then hu1, . . . , uri has finite index in Γ if and only if u1, . . . , ur are multiplicatively independent. A virtual basis of Γ is a set of multiplicatively independent elements of Γ which generate a subgroup of finite index in Γ. Siegel, Borel and Harish-Chandra showed that Z(U(ZG)) is finitely generated (Theorem 1.8.7) and its rank is known (Theorem 1.14.2). Therefore, one knows that a virtual basis of Z(U(ZG)) exists and one knows the number of elements in such a basis. In this chapter, we construct a virtual basis of Z(U(ZG)) for three classes of finite groups: for finite abelian groups; for finite abelian-by-supersolvable groups such that every cyclic subgroup of order not a divisor of 4 or 6, is subnormal in G; and for finite strongly monomial groups such that there exists a complete and non-redundant set of strong Shoda pairs (H,K) of G, with the property that each [H : K] is a power of a prime number.

4.1 abelian groups

In this section, we prove the Bass-Milnor Theorem (Theorem 1.12.1): the group generated by the Bass units is of finite index in U(ZG), for G a finite abelian group. We give a constructive proof that hence also provides insight in the techniques needed and difficulties one encounters in order to find finitely many generators for a subgroup of finite index. Additionally, we also discover a virtual basis consisting of Bass units.

101 central units

We assume that G is a finite abelian group. We know already that U(ZG) 1+k2+|G|−2c is finitely generated abelian (Theorem 1.8.7) and has rank r = 2 , where c is the number of cyclic subgroups of G and k2 is the number of elements of G of order 2 (Theorem 1.14.3). Bass took a concrete list of r Bass units of ZG for G cyclic and proved that they are multiplicatively independent. To do so, he used the Bass Independence Theorem which in turn uses the Franz Independence Lemma (see [Seh93, Lemma 11.3, Theorem 11.8] for details). Bass and Milnor proved, using K-Theory, that the group generated by the units of integral group rings of cyclic subgroups of G has finite index in U(ZG). However, their proof is not constructive. Alternatively, assume that Γ is a subgroup of finite index in a finitely generated abelian group Λ and that we know a subset X of Λ which generates a subgroup of finite index in Λ. Let Y be a subset of Γ. Then hY i has finite index in Γ if and only if for every x ∈ X there is a positive integer m such that xm ∈ hY i. In our proof we take Γ = U(ZG), Y the set of Bass units of ZG, Λ = U(O), where O is the unique maximal order of QG, and X the set of cyclotomic units of QG. By Theorem 1.6.4 of Perlis and Walker, QG is isomorphic to a direct product of cyclotomic fields. For simplicity, we consider this isomorphism as an equality Lk Lk QG = i=1 Q(ζni ). Then ZG is an order of QG and O = i=1 Z[ζni ] is the unique maximal order of QG. In particular, ZG ⊆ O. Moreover, Γ = U(ZG) has finite index in Λ = U(O) (see Lemma 1.8.6). Having in mind that the

cyclotomic units of Q(ζni ) generate a subgroup of ﬁnite index in U(Z[ζni ]) (see Section 1.11), we conclude that the set X of cyclotomic units of QG generates a subgroup of ﬁnite index in Λ. Thus the Bass-Milnor Theorem is equivalent to the following proposition.

Proposition 4.1.1 Lk Let G be a ﬁnite abelian group and let QG = i=1 Q(ζni ), the realization of the Perlis-Walker Theorem. Then for every cyclotomic unit u of QG there is a positive integer m such that um is a product of Bass units of ZG.

The proof of Proposition 4.1.1 (actually of Lemma 4.1.3) is constructive and avoids K-theory and independence arguments. This is the ﬁrst result of this section. The second result consists in giving a concrete virtual basis B formed by Bass units for U(ZG).

102 4.1 abelian groups

4.1.1 A new proof of the Bass-Milnor Theorem

Throughout the rest of the section G is a finite abelian group. In this section we prove Proposition 4.1.1. This provides a new proof of the Bass-Milnor Theorem which states that B1(G) has finite index in U(ZG). First of all we obtain a precise realization of the Perlis-Walker Theorem L kd which states that there is an isomorphism f : QG → d Q(ζd) , where kd denotes the number of cyclic subgroups of G of order d. This isomorphism is realized as follows. Let H = H(G) denote the set of subgroups H of G such that G/H is cyclic. For every subgroup H ∈ H, we fix a linear representation ρH of G with kernel H. We also denote by ρH the linear extension of ρH to QG. If d = [G : H] then ρH (QG) = Q(ζd), where ζd denotes a primitive d-th root of unity. Then M M f = ρH : QG −→ Q(ζ[G:H]) H∈H H∈H

is an isomorphism of algebras. This isomorphism is the same as the one of Perlis and Walker in Corollary 1.6.5, since kd equals the number of subgroups H ∈ H such that [G : H] = d. The following equalities are easy to check:

n−1 Y i n (1 − Xζn) = 1 − X , (12) i=0 m ρH (uk,m(g)) = ηk(ρH (g)) , (13) where H ∈ H, g ∈ G and km ≡ 1 mod |g|. Let ξ be a root of unity and assume that k is coprime to n and the order of ξ. If ξn 6= 1 then, using equation (12), we obtain

n−1 n−1 Y Y 1 − ξkζki Qn−1(1 − ξkζi ) 1 − ξkn η (ξζi ) = n = i=0 n = = η (ξn). k n i Qn−1 i 1 − ξn k i=0 i=0 1 − ξζn i=0 (1 − ξζn )

n j Otherwise, i.e. if ξ = 1, then ξζn = 1 for some j = 0, 1, . . . , n − 1. Then, using that k is coprime to n, we deduce that

n−1 n−1 k(i−j) Qn−1 ki Y i Y 1 − ζn i=1 (1 − ζn ) n ηk(ξζ ) = = = 1 = ηk(ξ ). n i−j Qn−1 i i=0 i=0,i6=j 1 − ζn i=1 (1 − ζn)

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This proves the following equality for every primitive l-th root of unity ξ:

n−1 Y i n ηk(ξζn) = ηk(ξ ) (gcd(k, nl) = 1). (14) i=0 Lemma 4.1.2 Let g ∈ G, H ∈ H and K be an arbitrary subgroup of G. Let h = |H ∩K|, t = [K : H∩K] and let k and m be positive integers such that gcd(k, t) = 1 and km ≡ 1 mod |gu| for every u ∈ K. Then

Y t mh ρH (uk,m(gu)) = ηk(ρH (g) ) . u∈K

Proof As H = ker(ρH ), if u runs through the elements of K then ρH (u) runs through the t-th roots of unity and each t-th root of unity is obtained as ρH (u) for precisely h elements u of K. Therefore !m Y Y ρH (uk,m(gu)) = ηk(ρH (g)ρH (u)) u∈K u∈K t−1 !mh Y i = ηk(ρH (g)ζt ) i=0 t mh = ηk(ρH (g) )

as desired. We have used equation (14) in the last equality. Proposition 4.1.1 is a consequence of the following stronger lemma. Lemma 4.1.3 Let H ∈ H with d = [G : H] and let k, j ∈ N be such that k is coprime to j d. Set η = ηk(ζd) and let B1,k(G) be the subgroup of U(ZG) generated by l the Bass cyclic units of the form uk,l(g) with g ∈ G and k ≡ 1 mod |g|. m Then there is a positive integer m and b ∈ B1,k(G) such that ρH (b) = η and ρK (b) = 1 for every K ∈ H \ {H}.

Proof Without loss of generality, we may assume that k is coprime to n = |G|. Indeed, by an easy Chinese Remainder argument there is an integer k0 coprime 0 j j to n such that k ≡ k mod d. Then clearly ηk(ζd) = ηk0 (ζd).

104 4.1 abelian groups

We argue by a double induction, ﬁrst on n and second on d. The cases n = 1 and d = 1 are trivial. We denote by P (G, H) the statement of the lemma for a ﬁnite abelian group G and an H ∈ H(G). Hence the induction hypothesis includes the following statements:

(IH1): P (M,Y ) holds for every proper subgroup M of G and any Y ∈ H(M).

(IH2): P (G, H1) holds for every H1 ∈ H(G) with [G : H1] < [G : H] = d.

We consider two cases, depending on whether j is coprime to d or not. Case 1: j is not coprime to d. Let p be a common prime divisor of d and j. j j/p Then H is contained in a subgroup S of G with [G : S] = p and ζd = ζd0 with 0 d = [S : H]. For every K ∈ H(G), let λK denote the restriction of ρK to QS. Clearly λK is the Q-linear extension of a linear representation of S with kernel S ∩K. Since S/(S ∩K) ' KS/K and KS/K is a subgroup of G/K we deduce that S/(S ∩ K) is cyclic. Thus K → S ∩ K deﬁnes a map H(G) → H(S). This map is surjective, but maybe not injective. Indeed, let K1 ∈ H(S). If K1 ∈ H(G) then clearly the map associates K1 with K1. Otherwise p divides [G : K1] and S/K1 is a cyclic subgroup of G/K1 of maximal order. This implies that G/K1 = S/K1 × L/K1 for some subgroup L of G containing K1 and so that [L : K1] = p. Then G/L ' S/K1, so that L ∈ H(G), and L ∩ S = K1. Therefore H(S) = {K ∩ S : K ∈ H(G)}. For every Y ∈ H(S) we choose a KY ∈ H(G) such that KY ∩ S = Y in such a way that KY = Y if Y ∈ H(G). Then M M λKY : QS → Q(ζ[S:Y ]) Y ∈H(S) Y ∈H(S)

is an algebra isomorphism. By the ﬁrst induction hypothesis (IH1) there is j/p m m an element b ∈ B1,k(S) such that λH (b) = ηk(ζd0 ) = η for some positive integer m and ρK (b) = λK (b) = 1 if K ∈ H(G) with K ∩ S 6= H. If K ∈ H(G) satisﬁes K ∩ S = H then either K = H or K = H1, where H1/H ' G/S is

the only subgroup of G/H of order p, since G/H is cyclic. Moreover ρH1 (b) is a product of cyclotomic units of Q(ζd), by equation (13). By the second induction hypothesis (IH2) there is c ∈ B1,k(G) such that ρK (c) = 1 for m1 every K ∈ H(G) \{H1} and ρH1 (c) = ρH1 (b) for some positive integer m1. m1 −1 mm1 m1 −1 Therefore ρH (b c ) = η and ρK (b c ) = 1 for every K ∈ H(G)\{H}. This ﬁnishes the proof for this case.

105 central units

j Case 2: j is coprime to d. Then G = ha, Hi and ρH (a) = ζd for some a ∈ G. As k is coprime to n, there is a positive integer m such that km ≡ 1 mod |au| for every u ∈ H. Hence m η = ρH (uk,m(a)), by equation (13). Let Y b = uk,m(ah). h∈H For every K ∈ H(G), set

0 dK = [G : K], dK = [G : ha, Ki], hK = |H ∩ K| and tK = [H : H ∩ K]. Then, by Lemma 4.1.2,

t d0 u ρ (b) = η (ρ (a)tK )m hK = η (ζ K K K )m hK , K k K k dK 0 for some integer uK coprime to dK . If tK dK is not coprime to dK then, by mK Case 1, there is bK ∈ B1,k(G) such that ρK (bK ) = ρK (b) for some integer

mK and ρK1 (bK ) = 1 for K1 ∈ H(G) \{K}. By (IH2), the same holds if dK < d. Let

0 0 H = {K ∈ H(G): tK dK is not coprime to dK or dK < d}.

0 For each K ∈ H ﬁx bK ∈ B1,k(G) and mK ∈ N as above and denote m1 = 0 lcm(mK : K ∈ H ) and ! − m1 Y mK m1 b1 = bK b . K∈H0

0 tK Then b1 ∈ B1,k(G), ρK (b1) = 1 if K ∈ H and ρK (b1) ∈ hηk(ρK (a) )i if 0 0 K ∈ H(G) \H . Observe that tH = dH = uH = 1 and dH = d and hence 0 tH H 6∈ H . Therefore, ρH (b1) ∈ hηi, because ηk(ρH (a) ) = η. To ﬁnish the proof we show that H0 = H(G) \{H}. Suppose the contrary, that is, assume 0 K ∈ H(G)\{H} with dK ≥ d and gcd(tK dK , dK ) = 1. The latter implies that 0 dK = 1, or equivalently G = ha, Ki, and tK = [KH : K] is coprime to dK = [G : K]. Consequently, tK = 1, or equivalently H ⊆ K. Hence, the assumption dK = [G : K] ≥ [G : H] = d implies that H = K, a contradiction. Note that the proof of Lemma 4.1.3 provides a recursive algorithm that for a cyclotomic unit η of QG as input, returns an integer m and an expression of ηm as a product of Bass units of ZG. For more details, see [JdRVG14].

106 4.1 abelian groups

As it was mentioned in the introduction of this section, the Bass-Milnor Theorem is equivalent to Proposition 4.1.1 and the well known fact that the cyclotomic units of Q(ζn) generate a subgroup of ﬁnite index of U(Z[ζn]) for every root of unity ζn. We include a proof for completeness.

Theorem 4.1.4 (Bass-Milnor)

If G is a ﬁnite abelian group then B1(G) has ﬁnite index in U(ZG).

L L Proof Let H = H(G), f = H∈H ρH : QG → H∈H Q(ζ[G:H]) and V be L the subgroup of H∈H U(Z(ζ[G:H])) generated by the cyclotomic units of QG, i.e. generated by the units that project on one component to a cyclotomic unit and project on all other components to 1. As the cyclotomic units of each ring Q(ζn) generate a subgroup of finite index in U(Z[ζn]), V has finite index L in H∈H U(Z(ζ[G:H])). Hence, by Lemma 4.1.3, f(B1(G)) has finite index L in H∈H U(Z(ζ[G:H])) and therefore B1(G) has finite index in U(ZG), since −1 L B1(G) ⊆ U(ZG) ⊆ f ( H∈H U(Z(ζ[G:H]))).

4.1.2 A virtual basis of Bass units

Bass proved that if G = hgi, a cyclic group of order n, and m is a multiple d n n of φ(n) then uk,m(g ) : 1 < k < 2d , (k, d ) = 1 is a virtual basis of U(ZG) [Bas66]. In this section we generalize this result and obtain a virtual basis consisting of Bass units for the unit group of the integral group ring of an arbitrary abelian group G. Moreover, the proof provides, for an arbitrary Bass unit b, an algorithm to express a power of b as a product of a trivial unit and powers of at most two units in this basis of Bass units.

Theorem 4.1.5 Let G be a ﬁnite abelian group. For every cyclic subgroup C of G, choose a generator aC of C and for every k coprime to the order of C, choose an m integer mk,C with k k,C ≡ 1 mod |C|. Then

ß |C| ™ u (a ): C cyclic subgroup of G, 1 < k < , gcd(k, |C|) = 1 k,mk,C C 2

is a virtual basis of U(ZG).

Proof The proof is based on the equalities from Lemma 1.12.2.

107 central units

Let C be the set of cyclic subgroups of G. By Theorem 4.1.4, B1(G) is a subgroup of ﬁnite index in U(ZG). Let t = φ(|G|). We ﬁrst prove that

ß |C| ™ B = u (a ): C ∈ C, 1 < k < , gcd(k, |C|) = 1 1 k,t C 2

generates a subgroup of finite index in U(ZG). To do so we “sieve” gradually the list of Bass units, keeping the property that the remaining Bass units still generate a subgroup of finite index in U(ZG), until the remaining Bass units are the elements of B1. By equation (1), to generate B1(G) it is enough to use the m Bass units of the form uk,m(g) with g ∈ G, 1 ≤ k < |g| and k ≡ 1 mod |g|. u v By equation (6), for every Bass unit uk,m(g) we have uk,m(g) = uk,t(g) for some positive integers u and v. Thus the Bass units of the form uk,t(g) with 1 ≤ k < |g| and gcd(k, |g|) = 1 generate a subgroup of finite index in U(ZG). By equations (2) and (3), we can reduce further the list of generators by taking only those with g = aC for some cyclic group C of G. By equations (4) and (5) we can exclude the Bass units with k = ±1 and still generate a subgroup of finite index in U(ZG) with the remaining elements. Finally, −1 k k −t uk,t(g) u|g|−k,t(g) = u|g|−1,t(g ) = (−g ) , by equations (3), (5) and (7). −1 Thus uk,t(g) u|g|−k,t(g) has finite order. Therefore the units uk,t(g) with |g| k > 2 can be excluded. The remaining units are exactly the elements of B1. Thus hB1i has finite index in U(ZG), as desired. C Let B = {uk,mk,C (aC ): C ∈ C, 1 < k < 2 , gcd(k, |C|) = 1}. Using equation (6) once more, we deduce that hBi has finite index in U(ZG), since so does hB1i. To finish the proof we need to show that the elements of B are multiplicatively independent. To do so, it is enough to show that the rank of U(ZG) coincides with the cardinality of B. For this, first observe that the cardinality P of B is d kdtd, where d runs through the divisors of |G|, kd is the number of cyclic subgroups of G of order d and td is the cardinality of {k : 1 < k < d φ(d) 2 , gcd(d, k) = 1}. Obviously t1 = t2 = 0 and td = 2 − 1 for every d > 2. 1+k +P h P Ä kdφ(d) ä 2 d d P 1+k2+|G|−2c Therefore, |B| = d>2 2 − kd = 2 − d kd = 2 , where hd denotes the number of elements of G of order d (so that h1 = 1 and h2 = k2) and c is the number of cyclic subgroups of G. By Theorem 1.14.3, this number coincides with the rank of U(ZG) and the proof is finished.

In Theorem 4.1.5 one can choose, for example, mk,C = φ(|G|), mk,C = φ(|C|) or mk,C = o|C|(k). Observe that the Bass Theorem is the specialization of

108 4.1 abelian groups

d Theorem 4.1.5 to G = hgi, ahgdi = g for d dividing |g|, and mk,hgdi a ﬁxed multiple of φ(|g|). The advantage of the new proof, with respect to the proofs of Bass and Bass-Milnor, is that it provides a way to express some power of any given Bass unit uk,m(g) as a product of a trivial unit and powers of at most 2 elements from ß |C| ™ B = u (a ): C cyclic subgroup of G, 1 < k < , gcd(k, |C|) = 1 k,mk,C C 2

for any given choice of generators aC of cyclic subgroups and integers mk,C as in Theorem 4.1.5. This is obtained as follows: calculate • n := |g|; C := hgi ; 0 0 0 k1 • k1 := the unique integer 0 ≤ k1 < n such that g = aC ; 0 0 • k0 := kk1 mod n; ® 0 0 0 1, if ki = ki; • for i = 0, 1 : k := min(k , n − k ); h := 0 i i i i ki aC , otherwise. M • M := lcm (m, mk0,C , mk1,C ); c := m ; Then, by equations (3), (6) and (8) from Lemma 1.12.2 we have

0 c k1 uk,m(g) = uk,M (aC ) −1 M −1 −M = u 0 (a )u 0 (a ) = u (a )h u (a ) h k0,M C k1,M C k0,M C 0 k1,M C 1 M M −1 M m − m = (h h ) u (a ) k0,C u (a ) k1,C . 0 1 k0,mk0,C C k1,mk1,C C We summarize this result in the following corollary. Corollary 4.1.6 Let G be a ﬁnite abelian group. For every cyclic subgroup C of G, choose a generator aC of C and for every k coprime to the order of C, choose an m integer mk,C with k k,C ≡ 1 mod |C|. Then ß |C| ™ u (a ): C cyclic subgroup of G, 1 < k < , gcd(k, |C|) = 1 k,mk,C C 2

is a virtual basis of U(ZG). Moreover, for any Bass unit uk,m(g) in ZG we have u (g)c = h u (a )n0 u (a )n1 , k,m k0,mk0,C C k1,mk1,C C

for C = hgi, an element h ∈ G and integers c, n0, n1, k0, k1 such that |C| ±k1 1 ≤ k0, k1 ≤ 2 , g = aC and k0 ≡ ±kk1 mod |C|.

109 central units

4.2 strongly monomial groups

Alternative to Theorem 1.14.2, we give a computation of the rank of the group of central units in the integral group ring ZG of a finite strongly monomial group G in terms of its strong Shoda pairs. Next, we construct generalized Bass units and show that the group they generate contains a subgroup of finite index in the central units of the integral group ring ZG for finite strongly monomial groups G. This generalizes a result of Eric Jespers and Michael M. Parmenter in [JP12, Corollary 2.3] on generators for central units of the integral group ring of a finite metabelian group. Theorem 4.2.1 Let G be a finite strongly monomial group. The rank of Z(U(ZG)) equals Ç å X φ([H : K]) − 1 , k(H,K)[NG(K): H] (H,K)

where (H,K) runs through a complete and non-redundant set of strong Shoda pairs of G, h is such that H = hh, Ki and

ß 1 if hhn ∈ K for some n ∈ N (K); k = G (H,K) 2 otherwise.

Proof By Proposition 1.7.13, one obtains the following description of the Wedderburn decomposition of QG: M M[G:N](Q(ζ[H:K]) ∗ N/H), (H,K) with (H,K) running through a complete and non-redundant set of strong Shoda pairs of G and N = NG(K). L N/H Since Z(QG) = (H,K) Q(ζ[H:K]) and by Lemmas 1.8.1 and 1.8.2, the L N/H order (H,K) Z[ζ[H:K]] is the unique maximal order of Z(QG). The rank N/H of Z(U(ZG)) is the sum of the ranks of the unit groups of Z[ζ[H:K]] , by Lemma 1.8.6. N/H Consider the center F = Q(ζ[H:K]) of the simple component

M[G:N](Q(ζ[H:K]) ∗ N/H).

110 4.2 strongly monomial groups

Clearly, [Q(ζ[H:K]): Q] φ([H : K]) [F : Q] = = . [Q(ζ[H:K]): F ] [N : H] Since F is a Galois extension of Q, we know that F is either totally real or totally complex. If F is totally real, then F is contained in the maximal −1 real subﬁeld Q(ζ[H:K] + ζ[H:K]) of Q(ζ[H:K]). This happens if and only if the Galois group N/H contains complex conjugation, which means that hhn ∈ K for some n ∈ N and h such that H = hh, Ki. Now using Dirichlet’s Unit Theorem 1.8.8, we obtain at once an appropriate rank computation. Let R be an associative ring with identity. Let x be a unit in R of ﬁnite order n. Let Cn = hgi, a cyclic group of order n. Then the map g 7→ x induces a ring homomorphism Z hgi → R. If k and m are positive integers with km ≡ 1 mod n, then the element 1 − km u (x) = (1 + x + ··· + xk−1)m + (1 + x + ··· + xn−1) k,m n

is a unit in R since it is the image of a Bass unit in Z hgi. In particular, if G is a ﬁnite group, M a normal subgroup of G, g ∈ G and k and m positive integers such that gcd(k, |g|) = 1 and km ≡ 1 mod |g|, then we have uk,m(1 − Mc + gMc) = 1 − Mc + uk,m(g)M.c

Observe that any element b = uk,m(1 − Mc + gMc) is an invertible element of ZG(1−Mc)+ZGMc. As this is an order in QG, there is a positive integer n such n that b ∈ U(ZG). Let nG,M denote the minimal positive integer satisfying this condition for all g ∈ G. Then we call the element

nG,M uk,m(1 − Mc + gMc) = uk,mnG,M (1 − Mc + gMc)

a generalized Bass unit based on g and M with parameters k and m. Note that we obtain the classical Bass units of ZG when M = 1. Before we prove that this construction of generalized Bass units yields a subgroup of ﬁnite index in Z(U(ZG)), we need the following lemma. Lemma 4.2.2

Let A1 and A2 be ﬁnite dimensional Q-algebras such that A1 ⊆ A2 and consider two orders O1 and O2 in A1 and A2 respectively. Then O2 ∩ A1 is an order in A1 and U(O2 ∩ A1) and U(O1) are commensurable.

111 central units

Proof Since O2 is a Z-module and A1 is a Q-algebra, clearly O2 ∩ A1 is a Z-module. Since O2 is a finitely generated Z-module and Z is Noetherian, all its submodules are finitely generated. Therefore O2 ∩ A1 is finitely generated. Also, since QO2 = A2, it follows that Q(O2 ∩ A1) = A1 and O2 ∩ A1 is an order in A1. The rest follows from Lemma 1.8.6. Theorem 4.2.3 Let G be a finite strongly monomial group. The group generated by the n 0 0 0 generalized Bass units b G,H with b = uk,m(1 − Hc + hHc) for a strong Shoda pair (H,K) of G and h ∈ H, contains a subgroup of finite index in Z(U(ZG)).

Proof Let (H,K) be a strong Shoda pair of G. Since H/H0 is abelian, it 0 follows from the Bass-Milnor Theorem 1.12.1 that B1(H/H ) is a subgroup of finite index in the group of (central) units of Z(H/H0) ' ZHHc0. A power of each Bass unit of Z(H/H0) is the natural image of a Bass unit in ZH. 0 0 Hence the group generated by units of the form b = uk,m(1 − Hc + hHc), with h ∈ H, is of finite index in U(Z(1 − Hc0) + ZHHc0). The group generated by the generalized Bass units bnG,M is still of finite index in U(Z(1 − Hc0) + ZHHc0). Let B1 denote this subgroup. Note that B1 is central in ZH. Since H0 ⊆ K ⊆ H, we know that ε(H,K) ∈ QHHc0 and hence Q(1 − ε(H,K)) + QHε(H,K) ⊆ Q(1 − Hc0) + QHHc0. Since Z(1−ε(H,K))+ZHε(H,K) is an order in Q(1−ε(H,K))+QHε(H,K) and Z(1 − Hc0) + ZHHc0 is an order in Q(1 − Hc0) + QHHc0, Lemma 4.2.2 implies that B2 = B1 ∩ (Z(1 − ε(H,K)) + ZHε(H,K)) is of finite index in U(Z(1 − ε(H,K)) + ZHε(H,K)). If α = 1 − ε(H,K) + βε(H,K) ∈ B2, with β ∈ ZH, and g ∈ NG(K), then αg = 1 − ε(H,K) + βgε(H,K). Since Z(1 − ε(H,K)) + ZHε(H,K) is a commutative ring, α and αg commute, and thus the product Q αg g∈NG(K) is independent of the order of its factors. Since U(ZHε(H,K)) is finitely generated, it is readily verified that {Q ug : u ∈ U( Hε(H,K))} is of g∈NG(K) Z finite index in U((ZHε(H,K))NG(K)/H ). Hence    Y g  B = α : α ∈ B2 ∩ (1 − ε(H,K) + ZHε(H,K))

g∈NG(K)  is a generating set for a subgroup which is of ﬁnite index in the unit group U(Z(1 − ε(H,K)) + (ZHε(H,K))NG(K)/H ).

112 4.3 abelian-by-supersolvable groups

Furthermore, if T is a transversal of NG(K) in G then ( ) Y C = γt : γ ∈ B t∈T

generates a subgroup of ﬁnite index in Z(U(Z(1−e(G, H, K))+ZGe(G, H, K))). To prove that the product in the deﬁnition of C is independent on the order of the product observe that if γ ∈ B and t1, t2 ∈ G then γ = 1 − ε(H,K) + t1 t2 γ1ε(H,K) for some γ1 ∈ ZH. If t1 6= t2 then ε(H,K) ε(H,K) = 0, because (H,K) is a strong Shoda pair. Using this it is easy to see that γt1 and γt2 commute. Take now an arbitrary central unit u in Z(U(ZG)). Then we can write this element as follows X Y u = ue(G, H, K) = (1 − e(G, H, K) + ue(G, H, K)), (H,K) (H,K) where (H,K) runs through a complete and non-redundant set of strong Shoda nG,H0 0 0 pairs of G. Note that conjugates of b = uk,mnG,H0 (1−Hc +hHc) are again of this form since conjugates of Bass units are again Bass units and because (Hg,Kg) is a strong Shoda pair of G if (H,K) is a strong Shoda pair of G for g ∈ G. Hence the result follows from the previous paragraph. Note that Theorem 4.2.3 extends the Bass-Milnor Theorem because for 0 0 abelian groups the generalized Bass units uk,m(1 − Hc + hHc) are precisely the Bass units.

4.3 abelian-by-supersolvable groups

In Section 4.2, we have proved that the group generated by the generalized Bass units contains a subgroup of finite index in Z(U(ZG)) for any arbitrary finite strongly monomial group G. Note that no multiplicatively independent set for such a subgroup was obtained. In this section we construct a virtual basis for the center of U(ZG) provided the finite group G is abelian-by-supersolvable and has the property that every cyclic subgroup of order not a divisor of 4 or 6 is subnormal in G. The basis elements are constructed as a (natural) product of conjugates of Bass cyclic units. For finite nilpotent groups G, constructions of central units of this type have earlier been considered by Jespers, Parmenter and Sehgal [JPS96]. Ferraz

113 central units and Sim´onconstructed in [FS08] a basis for the center of U(ZG) in case G is a metacyclic group of order pq, with p and q two distinct odd prime numbers.

4.3.1 Generalizing the Jespers-Parmenter-Sehgal Theorem

In this section we prove a generalization of a theorem of Jespers, Parmenter and Sehgal. We say that a subgroup H of a group G is subnormal in G if there is a finite chain of subgroups of the group, each one normal in the next, beginning at H and ending at G. For a finite abelian-by-supersolvable group G such that every cyclic subgroup of order not a divisor of 4 or 6, is subnormal in G, the detailed description of the primitive central idempotents of QG and the Bass-Milnor Theorem 1.12.1 allow us to show that B1(G) contains a subgroup of finite index in Z(U(ZG)). Furthermore, we obtain a description for the generators of this subgroup. In order to do this, we first need a new construction for central units based on Bass units in the integral group ring ZG. The idea originates from [JPS96], in which the authors constructed central units in ZG based on Bass units b ∈ ZG for finite nilpotent groups G. One denotes by Zi the i-th center, i.e. Z0 = 1 and Zi ¢ G is defined such that Zi/Zi−1 = Z(G/Zi−1). Since G is nilpotent, Zn = G for some n. For any g ∈ G and a Bass unit b based on g, put b(1) = b, and, for 2 ≤ i ≤ n, put

Y h b(i) = b(i−1). h∈Zi

By induction, b(i) is independent of the order of the conjugates in the product expression and b(i) is central in Z hZi, gi, since for every h ∈ Zi and for every i there exists x ∈ Zi−1 such that hg = xgh and hZi−1, gi ¢ hZi, gi. In particular, b(n) ∈ Z(U(ZG)). Note that the previous construction can be modified and improved by con- sidering the subnormal series hgi ¢ hZ1, gi ¢ ··· ¢ hZn, gi = G and taking in each step conjugates in a transversal for Zi in Zi−1. Then, the two constructions differ by a power. The constructions remain valid when starting with an arbitrary unit u in ZG with support in an abelian subgroup. We now generalize this construction to a bigger class of groups G. Through- out G will be a finite abelian-by-supersolvable group such that every cyclic subgroup of order not a divisor of 4 or 6, is subnormal in G. It is clear that this class of groups contains the finite nilpotent groups, the dihedral groups

114 4.3 abelian-by-supersolvable groups

n 2 −1 D2n = x, y : x = 1 = y , yxy = x and the generalized quaternion groups 2n 4 n 2 −1 −1 Q2n = x, y : x = 1 = y , x = y , y xy = x . Let u ∈ U(Z hgi), for g ∈ G of order not a divisor of 4 or 6. We consider a subnormal series N : N0 = hgi ¡ N1 ¡ N2 ¡ ··· ¡ Nm = G. Now deﬁne N c0 (u) = u and N Y N h ci (u) = ci−1(u) ,

h∈Ti where Ti is a transversal for Ni in Ni−1. We prove that this construction is well deﬁned by proving the following three properties. Lemma 4.3.1

Let g ∈ G, u ∈ U(Z hgi), N , Ni and Ti be as above. We have

N x (I) ci−1(u) ∈ ZNi−1, for all x ∈ Ni;

N x N (II) ci−1(u) = ci−1(u), for all x ∈ Ni−1;

N (III) ci (u) is independent on the choice of transversal Ti.

Proof It is easy to see that equation (II) implies (III). Hence it is suﬃcient to prove equations (I) and (II). We prove these by induction on i. First assume i = 1. Then equations (I) and (II) are trivial since the support of u is contained in hgi = N0 ¡ N1. N x Now assume the formulas hold for i − 1. Let x ∈ Ni. Then ci−1(u) = Q cN (u)hx. By the induction hypothesis we have that cN (u)h ∈ h∈Ti−1 i−2 i−2 N hx ZNi−2 and since Ni−2 ¡ Ni−1 ¡ Ni, also ci−2(u) ∈ ZNi−1, which proves (I). Now let x ∈ Ni−1. Then

N x Y N hx Y N h0 ci−1(u) = ci−2(u) = ci−2(u) . 0 h∈Ti−1 h ∈Ti−1x

Ti−1x remains a transversal for Ni−1 in Ni−2. Hence, by the induction hypo- N thesis on equation (III), the latter equals ci−1(u) and we have proved (II). By equations (I) and (II) we have that the construction is independent of N the order of the conjugates in the product expression. Furthermore, cm (u),

115 central units

the ﬁnal step in our construction, is a central unit in ZG, which we will simply denote by cN (u).

Theorem 4.3.2 Let G be a ﬁnite abelian-by-supersolvable group such that every cyclic subgroup of order not a divisor of 4 or 6, is subnormal in G. Then B1(G) contains a subgroup of ﬁnite index in Z(U(ZG)).

Proof We argue by induction on the order of the group G. For |G| = 1 the result is clear. So assume now that the result holds for groups of order strictly less than the order of G. Because of the Bass-Milnor Theorem 1.12.1, we can assume that G is non- abelian. Write QG = QG(1 − Gc0) ⊕ QGGc0. It is well known that QG(1 − Gc0) is a direct sum of non-commutative simple rings and QGGc0 ' Q(G/G0) is a commutative group ring. Hence, each z ∈ Z(U(ZG)) can be written as z = z0 + z00, with z0 ∈ Z(U(ZG(1 − Gc0))) and z00 ∈ U(ZGGc0). Note that z0z00 = 0 = z00z0. We will prove that some positive power of z is a product of Bass units. Since z is an arbitrary element of the finitely generated abelian group Z(U(ZG)), the result follows. First we focus on the commutative component. Since G/G0 is abelian, it 0 follows from the Bass-Milnor Theorem that B1(G/G ) is a subgroup of finite index in U(Z(G/G0)). A power of each Bass unit of Z(G/G0) is the natural 00m Qr image of a Bass unit of ZG. Hence, we get that z = i=1 bi for some positive integer m and some Bass units bi in ZG, where we denote the natural image of x ∈ ZG in Z(G/G0) by x. By Proposition 1.12.3, we know that uk,m(g) has finite order if and only if k ≡ ±1 mod |g|. In particular, there is a Bass unit based on g ∈ G of infinite order if and only if the order of g is not a divisor of 4 or 6. Hence we can assume that each bi is based on an element of order not a divisor of 4 or 6. By the assumptions on G, we can construct central units in ZG which project 0 Ni to some power of a bi in Z(G/G ). Indeed, each c (bi) is central in ZG, where Ni is a subnormal series from hgii to G when bi is based on gi. Since 0 Ni Z(G/G ) is commutative, the natural image of c (bi) is a power of bi, say mi m·lcm(mi:1≤i≤r) lcm(mi:1≤i≤r)/mi 00 Qr Ni bi . Hence z = i=1 c (bi) . Hence one may m0 0 00 Qs Nj assume there exists some positive integer m such that z = j=1 c (bj), where bj runs through a set of Bass units of ZG with possible repetition. 0 m Qs Nj −1 000 0 000 0 Therefore, z ( j=1 c (bj)) = z + Gc, with z ∈ Z(U(ZG(1 − Gc))).

116 4.3 abelian-by-supersolvable groups

Since G is abelian-by-supersolvable and hence also strongly monomial, we 0 L know that QG(1−Gc) = (H,K) QGe(G, H, K), where (H,K) runs through a complete and non-redundant set of strong Shoda pairs of G with QGe(G, H, K) not commutative. Note that in particular H 6= G for each such strong Shoda pair. Let (H,K) be a strong Shoda pair of G with H 6= G. Then it is also a strong Shoda pair of H and ε(H,K) is a primitive central idempotent of QH. Since |H| < |G|, the induction hypothesis yields that there exists a subgroup A1 in B1(H) such that A1 is of finite index in Z(U(ZH)). L Clearly, ZH ⊆ e ZHe, where e runs through all primitive central idempo- L tents of QH. As both ZH and e ZHe are Z-orders in QH, we have that L Z(U(ZH)) is of finite index in Z(U( e ZHe)). Hence, A1 is of finite index in L L Z(U( e ZHe)). Since Z(1 − ε(H,K)) + ZHε(H,K) ⊆ Z( e ZHe), we get that A = A(H,K) = A1 ∩ (Z(1 − ε(H,K)) + ZHε(H,K)) is of finite index in U(Z(1 − ε(H,K)) ⊕ ZHε(H,K)), and each element of A is a product of Bass units of ZH. From Proposition 1.7.13 we know that

QGe(G, H, K) ' M[G:NG(K)](QHε(H,K) ∗ (NG(K)/H)) and its center is isomorphic to (QHε(H,K))NG(K)/H , the fixed subfield of QHε(H,K) under the action of NG(K)/H. Since U(ZHε(H,K)) is a finitely generated abelian group, it is easy to verify that    Y n  u : u ∈ U(ZHε(H,K))

n∈NG(K)  is of ﬁnite index in U((ZHε(H,K))NG(K)/H ). Next note that if α = 1 − ε(H,K) + βε(H,K) ∈ A, with β ∈ ZH, then n n n α = 1 − ε(H,K) + β ε(H,K), for n ∈ NG(K). Hence, α and α commute and thus the product Q αn is independent of the order of its factors. n∈NG(K) It follows from the previous that    Y n  B = B(H,K) = α : α ∈ A ∩ (1 − ε(H,K) + ZHε(H,K))

n∈NG(K)  is a subgroup of ﬁnite index of U(Z(1 − ε(H,K)) + (ZHε(H,K))NG(K)/H ) and the elements of B are contained in B1(H).

117 central units

Let γ = 1 − ε(H,K) + δ ∈ B, with δ ∈ (ZHε(H,K))NG(K)/H . Let T be t t0 a right transversal of NG(K) in G. Since ε(H,K) ε(H,K) = 0 for different 0 t t0 Q t t, t ∈ T , we get that γ and γ commute and t∈T γ = 1 − e(G, H, K) + P t Q t t∈T δ ∈ 1 − e(G, H, K) + ZGe(G, H, K). Clearly, t∈T γ corresponds to a central matrix in QGe(G, H, K) with diagonal entry in (ZHε(H,K))NG(K)/H . Q t From the previous it follows that C = C(H,K) = { t∈T γ : γ ∈ B} is a subgroup of finite index in Z (U(Z(1 − e(G, H, K)) + ZGe(G, H, K))). As Q t each γ ∈ B is an element of B1(H), so is t∈T γ an element in B1(G). We can now finish the proof as follows. Write the central unit

X Y z000 + Gc0 = z000e(G, H, K) + Gc0 = (1 − e(G, H, K) + z000e(G, H, K)), (H,K) (H,K) where (H,K) runs through a complete and non-redundant set of strong Shoda P pairs of G so that QGe(G, H, K) is not commutative and (H,K) e(G, H, K) = 1 − Gc0. Because of the construction of C(H,K), there exists a positive integer 00 m00 so that (1 − e(G, H, K) + z000e(G, H, K))m ∈ C(H,K) for each (H,K). 00 Hence (z000 + Gc0)m , and thus also

00 s !m 0 00 00 m m Y Nj 000 0 m z = c (bj) (z + Gc) , j=1

is an element in B1(G).

Remark 4.3.3 Only one argument in the proof of Theorem 4.3.2 makes use of the assumption that every cyclic subgroup of order not a divisor of 4 or 6, is subnormal in G. It is needed to produce a central unit as a product of conjugates of a Bass unit b. For that we use the construction cN (b). It is not clear to us whether an alternative construction exists for other classes of groups, even for metacyclic groups this is unknown. At ﬁrst sight, it appears that one does not use the properties of abelian- by-supersolvable groups, except for the fact that these groups are strongly monomial and hence one knows an explicit description of the Wedderburn components. However, we cannot generalize the proof to strongly monomial groups since we use an induction hypothesis on subgroups and, unlike the class of abelian-by-supersolvable groups, the class of strongly monomial groups is not closed under subgroups.

118 4.3 abelian-by-supersolvable groups

Corollary 4.3.4 Let G be a finite abelian-by-supersolvable group such that every cyclic subgroup of order not a divisor of 4 or 6, is subnormal in G. For each such cyclic subgroup hgi, fix a subnormal series Ng from hgi to G. The group Ng c (bg): bg a Bass unit based on g, g ∈ G is of finite index in Z(U(ZG)).

Proof Because Z(U(ZG)) is ﬁnitely generated by Theorem 1.8.7 and using Theorem 4.3.2, it is suﬃcient to show that if u = b1b2 ··· bm ∈ Z(U(ZG)), with each bi a Bass unit based on gi ∈ G, then there exists a positive integer l Ng l so that u is a product of c (bg)’s, with bg a Bass unit based on g ∈ G. In order to prove this, for each primitive central idempotent e of QG, write

QGe = Mne (De), with ne a positive integer and De a division algebra. If Oe Q is an order in De, then we have that U(ZG) ∩ e GLne (Oe) is of ﬁnite index in U(ZG) (cf. Lemma 1.8.6). It is easy to verify that the central matrices in

SLne (Oe) have ﬁnite order. Now, let u = b1b2 ··· bm ∈ Z(U(ZG)), with each bi a Bass unit based on gi ∈ 0 m0 0 m Qm Ng G. Then there exists a positive integer m such that u , i=1 c i (bi) ∈ Q Ngi e GLne (Oe). Let ki be the positive integer so that each c (bi) is a product Ng ki of ki conjugates of bi. Then c i (bi)e and bi e have the same reduced norm. Hence, m 0 Y 0 km Ngi −m k/ki u c (bi) e ∈ SLne (Oe) ∩ Z(GLne (Oe)), i=1 0 0 km Qm Ng −m k/ki for k = lcm(ki : 1 ≤ i ≤ m) and thus u i=1 c i (bi) e is an element

of ﬁnite order in Z(GLne (Oe)). Consequently,

00 m !m 0 0 km Y Ng −m k/ki u c i (bi) = 1 i=1 for some positive integer m00, i.e.

0 00 km m Ng u ∈ c (bg): bg a Bass unit based on g, g ∈ G . For ﬁnite nilpotent groups of class n, we can always take the subnormal Ng series Ng : hgi ¢ hZ1, gi ¢ ··· ¢ hZn, gi = G. Since both constructions c (b) and b(n) only diﬀer by a power, we can deduce the Jespers-Parmenter-Sehgal result.

119 central units

Corollary 4.3.5 (Jespers-Parmenter-Sehgal) Let G be a ﬁnite nilpotent group of class n. The group

b(n) : b ∈ B1(G)

is of ﬁnite index in Z(U(ZG)).

4.3.2 Reducing to a basis of products of Bass units

In this section, we reduce the generating set of a subgroup of finite index in Z(U(ZG)) from Corollary 4.3.4 to a virtual basis of Z(U(ZG)), for G a finite abelian-by-supersolvable group G such that every cyclic subgroup of order not a divisor of 4 or 6, is subnormal in G. First, we need some properties of our construction of central units. Lemma 4.3.6 Let G be a finite group. Let u, v be units in Z hgi for g ∈ G and let N be a subnormal series N0 = hgi ¡ N1 ¡ ··· ¡ Nm = G. Assume that h is a group element of G and denote by N h the h-conjugate of the series N , h h h h h i.e. N : N0 = g ¡ N1 ¡ ··· ¡ Nm = G. Then (A) cN (uv) = cN (u)cN (v); and

h (B) cN (uh) = cN (u).

N N N Proof Let u, v ∈ Z hgi. Then clearly c0 (uv) = uv = c0 (u)c0 (v). By an induction argument on i, we now get that

N Y N x Y N x N x N N ci (uv) = ci−1(uv) = ci−1(u) ci−1(v) = ci (u)ci (v),

x∈Ti x∈Ti N x N x for i ≥ 1, since ci−1(u) and ci−1(v) commute by properties (I) and (II). This proves equation (A). N h h N h Let u ∈ Z hgi and h ∈ G. We prove that ci (u ) = ci (u) by induction N h h h N h on i. For i = 0, we have c0 (u ) = u = c0 (u) . Let i ≥ 1, now by the induction hypothesis

N h h Y N h h x Y N hx Y N yh N h ci (u ) = ci−1(u ) = ci−1(u) = ci−1(u) = ci (u) .

h h y∈Ti x∈Ti x∈Ti

120 4.3 abelian-by-supersolvable groups

Let G be a group and g ∈ G. Deﬁne

l Sg = {l ∈ U(Z/|g|Z): g is conjugate with g in G}.

In other words, Sg is the image of the homomorphism

NG(hgi) → U(Z/|g|Z): h 7→ lh,

h l where lh is the unique element of U(Z/|g|Z) such that g = g h . The kernel of this homomorphism is CenG(g). We denote Sg = hSg, −1i and we always assume that transversals of Sg in U(Z/|g|Z) contain the identity 1. Now we give a generalization of Theorem 4.1.5. For this, we need that a ﬁnite group G, satisfying the assumptions of Theorem 4.3.2, is almost nilpotent. Lemma 4.3.7

Let Y = {g ∈ G : there exists a subnormal series Ng from hgi to G}. Then hY i is a nilpotent group.

Proof To prove that hY i is nilpotent, it is suﬃcient to prove that the p- elements of hY i form a subgroup, i.e. that products of such elements are again p-elements. For this, it is suﬃcient to show that all generators of hY i of order a power of p commute with all p0-elements in hY i. Indeed, assume that all p-elements of hY i commute with all p0-elements. Let y1, . . . , yk be p-elements, with p prime. Write y1 ··· yk = zpzp0 , with zp 0 a p-element and zp0 a p -element. Then, by the assumption, zp0 is central in hy1, ··· , yki. Since the latter group is solvable, we know from the Hall result that the group hy1, ··· , yki contains a subgroup P such that its index is not divisible by p and a subgroup P 0 such that the only prime dividing its index is p. From the assumptions, we know that P 0 is central in this group and thus 0 0 it follows that hy1, ··· , yki = P × P . Hence zp0 ∈ P and y1, . . . , yk ∈ P . This implies zp0 = 1 and thus y1 ··· yk is a p-element, as desired. We prove the result by induction on the order of G. If |G| = 1 then the result is obvious. Note that subgroups of G satisfy the same assumption. So we assume that the result holds for groups of order less than |G| and also G = hY i. We need to prove that all generators of G = hY i that are p-elements commute with all p0-elements of G. Let x be such a p-element and y such a q-element. If hx, yi is a proper subgroup of G then the result holds by the induction hypothesis. So we may assume that G = hx, yi. We need to prove

121 central units

that x and y commute. Since G is abelian-by-supersolvable, there exists a normal series 1 ¡ A = B0 ¡ B1 ¡ ··· ¡ Bk = G, with A abelian and each Bi+1/Bi a cyclic group of prime order. Clearly, Bk/Bk−1 = hx, yi. Hence without loss of generality we may assume that Bk/Bk−1 = hyi and thus Bk/Bk−1 is a cyclic group of order q. Obviously, Bk−1 is a proper subgroup and thus also ¨ yi q ∂ H = x, x , y | i ∈ N ⊆ Bk−1 is a proper subgroup of G and thus by the induction hypothesis it is nilpotent. So, H is a direct product of its Sylow subgroups. One of its Sylow subgroups is a Sylow p-subgroup, say P , with x ∈ P . Since this is an invariant subgroup of G, we get that hP, yi = P o hyi. If this is a direct product then x ∈ P and y commute, as desired. So suppose that x and y do not commute. Then consider a series

hyi ¡ N1 ¡ ··· ¡ Nl = P o hyi .

Let i be the smallest index such that Ni contains an element, say z, that does not commute with y. We may assume that z ∈ P and thus z is a p-element. −1 −1 y Let 1 6= t = z y zy ∈ Ni−1. In particular, t and y commute. Then z = zt q and thus z = zy = ztq. Now tq = 1. As t is a p-element this implies t = 1, a contradiction. Note that necessarily hY i is a characteristic subgroup of G. Hence, by Lemma 4.3.7, for each g ∈ Y , there exists a series

hgi ¢ hZ1, gi ¢ hZ2, gi ¢ ··· ¢ hZm, gi = hY i ¢ G, with Zi the i-th center of hY i. Since each hZi, gi is normalized by NG(hgi), we can assume that for each g ∈ Y , there exists a subnormal series Ng from hgi to G, which is normalized by NG(hgi). Theorem 4.3.8 Let G be a ﬁnite abelian-by-supersolvable group such that every cyclic subgroup of order not a divisor of 4 or 6, is subnormal in G. Let R denote a set of representatives of Q-classes of G. For g ∈ R, choose a transversal Tg of Sg in U(Z/|g|Z) containing 1 and for every k ∈ Tg \{1}, choose an m integer mk,g with k k,g ≡ 1 mod |g|. For every g ∈ R, of order not a divisor of 4 or 6, choose a subnormal series Ng from hgi to G, which is normalized by NG(hgi). The set

Ng c (uk,mk,g (g)) : g ∈ R, k ∈ Tg \{1}

is a virtual basis of Z(U(ZG)).

122 4.3 abelian-by-supersolvable groups

Proof For every g ∈ R of order not a divisor of 4 or 6, we choose a subnormal series Ng from hgi to G, which is normalized by NG(hgi). For each h ∈ G of order not a divisor of 4 or 6, we agree to choose the subnormal series Nh to −1 i be the x-conjugate of Ng when h = x g x, with g ∈ R and i coprime to the order of g. By Corollary 4.3.4, the set

Nh m B1 = {c (uk,m(h)) : h ∈ G, k, m ∈ N, k ≡ 1 mod |h|}

generates a subgroup of finite index in Z(U(ZG)). Let t = φ(|G|). We first prove that Ng B2 = c (uk,t(g)) : g ∈ R, k ∈ Tg \{1} generates a subgroup of finite index in Z(U(ZG)). To do so we sieve gradually the list of units in B1, keeping the property that the remaining units still generate a subgroup of finite index in Z(U(ZG)), until the remaining units are the elements of B2. For this, we use the equations from Lemma 1.12.2. By equation (1), to generate B1 it is enough to use the Bass units of the m form uk,m(h) with h ∈ G, 1 ≤ k < |h| and k ≡ 1 mod |h|. Hence one can assume that k ∈ U(Z/|h|Z). i j By equation (6), for every Bass unit uk,m(h) we have uk,m(h) = uk,t(h) for some positive integers i and j. Thus, by equation (A), units of the N form c h (uk,t(h)) with k ∈ U(Z/|h|Z) generate a subgroup of finite index in Z(U(ZG)). By the definition of a Q-class, we know that each h ∈ G is conjugate to some gi, for g ∈ R and gcd(i, |g|) = 1. Hence, by equations (3), (A) and (B), we can reduce further the list of generators by taking only Bass units based on elements of R. By equation (4), we can exclude k = 1 and still generate a subgroup of finite index in Z(U(ZG)). Let g ∈ G be of order n. We claim that if l ∈ Sg and k ∈ U(Z/nZ) then Ng k k k −lkt c (ul,t(g )) has finite order. As un−l,t(g ) = ul,t(g )g , by (8), we may assume without loss of generality that l ∈ Sg. By equations (3), (A) and (B) we have

Ng k Ng k Ng kl Ng k Ng k c (uli+1,t(g )) = c (ul,t(g ))c (uli,t(g )) = c (ul,t(g ))c (uli,t(g )).

Then, arguing inductively we deduce that

Ng k i Ng k c (ul,t(g )) = c (uli,t(g )),

123 central units

Ng k t Ng k Ng k and in particular c (ul,t(g )) = c (ult,t(g )) = c (u1,t(g )) = 1, by equations (1) and (4). This proves the claim. With g and n as above, every element of U(Z/nZ) is of the form kl with k k ∈ Tg and l ∈ Sg. Using (3) again we have ukl,t(g) = uk,t(g)ul,t(g ). By the Ng k previous paragraph, c (ul,t(g )) has ﬁnite order. Hence we can reduce the generating system and take only k ∈ Tg \{1}.

The remaining units are exactly the elements of B2. Thus hB2i has finite index in Z(U(ZG)), as desired. Ng Let B = {c (uk,mk,C (g)) : g ∈ R, k ∈ Tg \{1}}. Using (6) once more, we deduce that hBi has finite index in Z(U(ZG)), since hB2i does. To finish the proof we need to prove that the elements of B are multiplicatively independent. To do so, it is enough to show that the rank of Z(U(ZG)) coincides with the cardinality of B. It is easy to see that P |B| = g∈R|Tg| − |R| and |R| equals the number of Q-classes. By construction, [U(Z/|g|Z): Sg] equals the number of conjugacy classes contained in the Q-class of g. Furthermore, [Sg : Sg] = 1 when g is conjugated to −1 −1 g and [Sg : Sg] = 2 when g is not conjugated to g . Therefore |Tg| = [U(Z/|g|Z): Sg] is exactly the number of R-classes contained in the Q-class of g. Hence |B| equals the number of R-classes minus the number of Q-classes in G. By Theorems 1.14.1 and 1.14.2, this number coincides with the rank of Z(U(ZG)) and the proof is finished.

4.4 another class within the strongly monomial groups

In Section 4.3, we obtained an explicit description of a virtual basis of the central units Z(U(ZG)) when G is a finite abelian-by-supersolvable group such that every cyclic subgroup of order not a divisor of 4 or 6 is subnormal in G. Note that the latter does not apply to all finite split metacyclic groups Cm ok Cn, for example if n is a prime number and Cm ok Cn is not abelian then Cn is not subnormal in Cm ok Cn. On the other hand, Ferraz and Simón did construct in [FS08] a virtual basis of Z(U(Z(Cq o1 Cp))) for p and q odd and different prime numbers. We extend these results on the construction of a virtual basis of Z(U(ZG)) to a class of finite strongly monomial groups containing the metacyclic groups G = Cqm o1 Cpn , with p and q different prime numbers. We focus on strongly monomial groups G such that there is a complete and non-redundant set of strong Shoda pairs (H,K) of G with the property that

124 4.4 another class within the strongly monomial groups

[H : K] is a prime power. For this class of strongly monomial groups, we construct a virtual basis for the group Z(U(ZG)). The construction of units in the basis is inspired by the construction of Bass units. Since the Wedderburn decomposition of QG is well described, we want to exploit this knowledge to describe a virtual basis of Z(U(ZG)) in terms of cyclotomic units of QG. In order to compute a virtual basis of Z(U(ZG)), we will “cover” the central integral units in the different simple components by using generalized Bass units. This will lead to a final description of the central units up to finite index. Indeed, take an arbitrary central unit u in Z(U(ZG)). Then we can write this element as follows X Y u = ue(G, H, K) = (1 − e(G, H, K) + ue(G, H, K)), (H,K) (H,K) where (H,K) runs through a complete and non-redundant set of strong Shoda pairs of G. Hence it is necessary and sufficient to construct a set of multiplicatively independent units in the center of each order ZGe(G, H, K) + Z(1 − e(G, H, K)). NG(K)/H The centers of ZGe(G, H, K) + Z(1 − e(G, H, K)) and Z[ζ[H:K]] + Z(1 − e(G, H, K)) are both orders in the center of the algebra QGe(G, H, K) + Q(1 − e(G, H, K)) and therefore their unit groups are commensurable. Hence, NG(K)/H we are interested in the units of Z[ζ[H:K]] and furthermore in the NG(K)/H units of ZG projecting to units in Z[ζ[H:K]] and trivially to the other components. r We define σr ∈ Aut(hζpn i) ' Gal(Q(ζpn )/Q) by σr(ζpn ) = ζpn . For a Q subgroup A of Aut(hζpn i) and u ∈ Q(ζpn ), we define πA(u) to be σ∈A σ(u). Since, by assumption, [H : K] equals a prime power, say pn, it is well known that Aut(H/K) is cyclic, unless p = 2 and n ≥ 3 in which case Aut(H/K) = hσ5i × hσ−1i. We make abuse of notation to identify NG(K)/H to the Galois NG(K)/H group of Q(ζpn )/Q(ζpn ) and with a subgroup of U(Z/[H : K]Z). The NG(K)/H Galois group Gal(Q(ζpn )/Q(ζpn ) ) is a subgroup of Aut(H/K) and it follows that NG(K)/H is either hσri for some r or hσri × hσ−1i for some r ≡ 1

mod 4. We simply denote πNG(K)/H by π and have

Y Y π(u) = σ(u) = ui

σ∈NG(K)/H i∈NG(K)/H

for u ∈ Q(ζpn ). We will need the following lemma.

125 central units

Lemma 4.4.1

Let A be a subgroup of Aut(hζpn i). Let I be a set of coset representatives n of U(Z/p Z) modulo hA, σ−1i containing 1. The set

{πA (ηk(ζpn )) : k ∈ I \{1}}

A is a virtual basis of U Z[ζpn ] .

Proof Assume A = hσri or A = hσri×hσ−1i. In both cases, we set l = | hσri |. The arguments at the end of the proof of Theorem 4.2.1 show that the unit n−1 A p (p−1) group U Z[ζpn ] has rank ld − 1 = |I| − 1, where d = 2 if −1 6∈ hri and d = 1 otherwise. Moreover ld = | hA, σ−1i |. By Theorem 1.11.2, the cyclotomic units of the form ηk(ζpn ) generate a A subgroup of ﬁnite index in U(Z[ζpn ]). Therefore, for every unit u of Z[ζpn ] , m h m|A| Q n u = i=1 ηki (ζp ) for some integers m, k1, . . . , kh. Then, u = πA(u) = h Q n i=1 πA(ηki (ζp )). Hence, it is clear that

n {πA(ηk(ζpn )) : k ∈ U(Z/p Z)}

A generates a subgroup of ﬁnite index in U Z[ζpn ] . Throughout the proof we use the equalities from Lemma 1.11.1. First consider the case when d = 1 (i.e. −1 ∈ hri). Because of equation (2), we have i ηrti(ζpn ) = ηi(ζpn )ηrt (ζpn ),

i irt for i ∈ I and 0 ≤ t ≤ l − 1. Note that πA(ηrt (ζpn )) = πA(ηrt (ζpn )), for 0 ≤ t ≤ l − 1. Again by (2), we deduce that

i 2 i irt i (πA(ηrt (ζpn ))) = πA(ηrt (ζpn ))πA(ηrt (ζpn )) = πA(ηr2t (ζpn )),

i and hence it follows by induction and equations (1) and (3) that πA(ηrt (ζpn )) has ﬁnite order. Hence {πA(ηi(ζpn )) : i ∈ I} A still generates a subgroup of ﬁnite index in U Z[ζpn ] . Now consider the case when d = 2 (i.e. −1 6∈ hri). Let J = I ∪−I. Then J is a set of coset representatives of U(Z/pnZ) modulo hri. By the same arguments as above, we can deduce that

{πA(ηk(ζpn )) : k ∈ J}

126 4.4 another class within the strongly monomial groups

A generates a subgroup of ﬁnite index in U Z[ζpn ] . If i ∈ I, then, by equation −i −i (4), we have that πA(η−i(ζpn )) = πA(−ζpn )πA(ηi(ζpn )) and πA(−ζpn ) is of ﬁnite order. Thus

{πA(ηi(ζpn ))} : i ∈ I}

A still generates a subgroup of ﬁnite index in U Z[ζpn ] . Now, in both cases, by equation (3), we can exclude i = 1. And since the size now coincides with the rank, the set

{πA(ηk(ζpn )) : k ∈ I \{1}}

A is a virtual basis of U Z[ζpn ] .

Because of the natural isomorphism Q(H/K) ' QHK“, the following lemma is a translation of Lemma 4.1.2. It gives a direct link between generalized Bass units and cyclotomic units.

Lemma 4.4.2 Let H be a ﬁnite group, K a subgroup of H and g ∈ H such that H/K = hgKi. Put H = {L ≤ H : K ≤ L}. For every L ∈ H, ﬁx a linear representation ρL of H with kernel L. Assume that L ∈ H and M is an arbitrary subgroup of H. Let l = |L ∩ M|, t = [M : L ∩ M] and let k and m be positive integers such that gcd(k, t) = 1 and km ≡ 1 mod |gu| for every u ∈ M. Then

Y t lmnH,K ρL(uk,mnH,K (guK“ + 1 − K“)) = ηk(ρL(g) ) . u∈M

Let H be a ﬁnite group and K a subgroup of H such that H/K = hgKi is a cyclic group of order pn. It follows that the subgroups of H/K form a chain, ¨ pn−j ∂ hence H = {L : K ≤ L ≤ H} = {Hj = g ,K : 0 ≤ j ≤ n}. This is a crucial property to prove the next result. Let k be a positive integer coprime with p and let r be an arbitrary integer. For every 0 ≤ j ≤ s ≤ n, we construct recursively the following products of generalized Bass units of ZH:

s cs(H, K, k, r) = 1,

127 central units and, for 0 ≤ j ≤ s − 1, Ñ éps−j−1 s Y rpn−s cj (H, K, k, r) = uk,opn (k)nH,K (g hK“ + 1 − K“) h∈Hj

Ñ s−1 é j−1 ! Y s −1 Y s+l−j −1 cl (H, K, k, r) cl (H, K, k, r) . l=j+1 l=0 Here, we agree that by definition an empty product equals 1. The idea originates from [JdRVG14, Proposition 4.2] for finite elementary p-groups, where the result arises naturally. Proposition 4.4.3 Let H be a finite group and K a subgroup of H such that H/K = hgKi n ¨ pn−j ∂ is cyclic of order p . Let H = {Hj = g ,K : 0 ≤ j ≤ n}. Let k be a positive integer coprime with p and let r be an arbitrary integer. Then

® s−1 η (ζr )opn (k)p nH,K if j = j ; ρ (cs(H, K, k, r)) = k ps−j 1 (15) Hj1 j 1 if j 6= j1,

for every 0 ≤ j, j1 ≤ s ≤ n.

Proof We use a double induction, first on s and, for a fixed s, on s − j. The minimal cases (s = 1 and s = j) are obvious, so assume that s > 1, j < s and the result holds for s1 < s and 0 ≤ j, j1 ≤ s1, and for s fixed and j1 > j. n−s Let u(k) = Q u (grp hK + 1 − K). By Lemma 4.4.2, if h∈Hj k,opn (k)nH,K “ “ 0 ≤ i ≤ j, then

n−s j−i i j−i i rp p opn (k)nH,K p rp opn (k)nH,K p ρHi (u(k)) = ηk(ρHi (g ) ) = ηk(ζps−i ) , (16) and if j ≤ i ≤ n then

n−s j j rp opn (k)nH,K p r opn (k)nH,K p ρHi (u(k)) = ηk(ρHi (g )) = ηk(ζps−i ) . (17) By (16) and the induction hypothesis on s, we have ρ (cs+i−j(H, K, k, r)) Hj1 i s+i−j−1 s−j−1 ® r opn (k)nH,K p p ηk(ζ s−j ) = ρH (u(k)) if j1 = i < j; = p j1 (18) 1 if j1 6= i < j.

128 4.4 another class within the strongly monomial groups

By (17) and the induction hypothesis on s − j, we have

ρ (cs(H, K, k, r)) Hj1 i s−1 s−j−1 ® r opn (k)nH,K p p ηk(ζ s−i ) = ρH (u(k)) if j < i = j1; = p j1 (19) 1 if j < i 6= j1.

Now, combining (16), (18) and (19), we have

ρ (cs(H, K, k, r)) Hj1 j  s−j−1 s−1 ρ (u(k))p = η (ζr )opn (k)nH,K p if j = j ;  Hj k ps−j 1  s−j−1 = ρ (u(k))p ρ (cs (H, K, k, r))−1 = 1 if j < j ; Hj1 Hj1 j1 1 s−j−1  ρ (u(k))p ρ (cs+j1−j(H, K, k, r))−1 = 1 if j > j , Hj1 Hj1 j1 1 as desired. We state now our main theorem on central units of this section. Observe that we are again identifying NG(K)/H with a subgroup of U(Z/[H : K]Z) for a strong Shoda pair (H,K) of G. Theorem 4.4.4 Let G be a ﬁnite strongly monomial group such that there exists a complete and non-redundant set S of strong Shoda pairs (H,K) of G, with the property that each [H : K] is a prime power. For every (H,K) ∈ S, let TK be a right transversal of NG(K) in G, let I(H,K) be a set of representatives of U(Z/[H : K]Z) modulo hNG(K)/H, −1i containing 1 and n(H,K) let [H : K] = p(H,K) , with p(H,K) a prime number. The set    Y Y n(H,K) t  c0 (H, K, k, x) :(H,K) ∈ S, k ∈ I(H,K) \{1}

t∈TK x∈NG(K)/H 

is a virtual basis of Z(U(ZG)).

Proof Fix (H,K) ∈ S, N = NG(K), T = TK , I = I(H,K) and n = n(H,K). It is suﬃcient to prove that   Y Y n t  c0 (H, K, k, x) : k ∈ I \{1} t∈T x∈N/H 

129 central units

is a virtual basis of the center of 1 − e(G, H, K) + U(ZGe(G, H, K)). To prove this, we may assume without loss of generality that K is normal in G, or equivalently N = G. Indeed, assume we can compute a virtual basis {u1, . . . , us} of the center of 1−ε(H,K)+U(ZNε(H,K)). Each ui is of the form −1 0 1 − ε(H,K) + viε(H,K) for some vi ∈ ZN and ui = 1 − ε(H,K) + viε(H,K) 0 Q t P t t for some vi ∈ ZN. Then, wi = t∈T ui = 1−e(G, H, K)+ t∈T vi ε(H,K) is a unit in the center of 1−e(G, H, K)+ZGe(G, H, K) since the ε(H,K)t are mutually orthogonal idempotents and they also are orthogonal to 1 − e(G, H, K). Then w1, . . . , ws are independent because so are u1, . . . , us and they form a virtual basis of the center of 1 − e(G, H, K) + ZGe(G, H, K). From now on we assume that K is normal in G and [H : K] = pn with p a prime number. Thus N = G and T = {1}. We have to prove that    Y n  c0 (H, K, k, x): k ∈ I \{1} x∈G/H 

is a virtual basis of the center of 1 − e(G, H, K) + U(ZGe(G, H, K)). Assume ﬁrst that H = G. Then QGe(G, G, K) ' QGK“ ' Q(ζpn ). Con- sider the algebra QGe(G, G, K)+Q(1−e(G, G, K)) as a subalgebra of QGK“ + n Q(1 − K“). By Proposition 4.4.3, the elements c0 (G, K, k, 1) project to a cyclo- n−1 o n (k)p nG,K tomic unit ηk(ζpn ) p in the simple component Q(ζpn ) and trivially in all other components. Since we know that the set {ηk(ζpn ): k ∈ I \{1}} is a virtual basis of U(Z[ζpn ]) (cf. Lemma 4.4.1), it follows that

n {c0 (G, K, k, 1) : k ∈ I \{1}}

is a virtual basis of 1 − e(G, G, K) + U(ZGe(G, G, K)). Now, assume that H 6= G and consider the non-commutative simple component QGe(G, H, K) ' QHε(H,K) ∗ G/H with center (QHε(H,K))G/H ' G/H G/H Q(ζpn ) ⊆ QHK“. Consider the commutative algebra (QHε(H,K)) + Q(1 − ε(H,K)) as a subalgebra of QHK“ + Q(1 − K“). Since H/K is a cyclic p-group, G/H = hσri or G/H = hσri × hσ−1i for some r. Say | hσri | = l. By Lemma 4.4.1, the set {π (ηk(ζpn )) : k ∈ I \{1}} is a virtual basis of G/H U Z[ζpn ] . If G/H is cyclic, by Proposition 4.4.3, the elements

n n n l−1 c0 (H, K, k, 1)c0 (H, K, k, r) ··· c0 (H, K, k, r )

130 4.4 another class within the strongly monomial groups

n−1 o n (k)p nH,K G/H project to π(ηk(ζpn )) p in the component Q(ζpn ) and trivially in all other components of QH. Hence the set

n n n l−1 {c0 (H, K, k, 1)c0 (H, K, k, r) ··· c0 (H, K, k, r ): k ∈ I \{1}}

is a virtual basis of Z(U(ZGe(G, H, K) + Z(1 − e(G, H, K)))). If G/H is not cyclic, then the elements

l−1 1 Y Y n i j c0 (H, K, k, r (−1) ) i=0 j=0

G/H project to a power of π(ηk(ζpn )) in the component Q(ζpn ) and trivially in all other components of QH. Hence also in this case we ﬁnd a set

(l−1 1 ) Y Y n i j c0 (H, K, k, r (−1) ): k ∈ I \{1} , i=0 j=0 which is a virtual basis of Z(U(ZGe(G, H, K) + Z(1 − e(G, H, K)))). We apply our results to construct a virtual basis of the group Z(U(ZG)), for the metacyclic groups of the form Cqm o1 Cpn , for two diﬀerent prime numbers p and q. This generalizes results from Ferraz and Sim´onwhere the case m = n = 1 is handled [FS08]. Corollary 4.4.5

Let p and q be diﬀerent prime numbers. Let G = Cqm o1 Cpn be a ﬁnite metacyclic group with Cpn = hbi and Cqm = hai. Let r be such that b r a = a . For each j = 1, . . . , m, let Ij be a set of coset representatives of U(Z/qjZ) modulo hr, −1i. If p = 2, then

¶ i ¨ 2i ∂ i−1 © U = c0(G, a, b , k, 1) : 1 < k < 2 , 2 - k, i = 2, . . . , n ß S m m m 2n−1 cm−j(hai , 1, k, 1)cm−j(hai , 1, k, r) ··· cm−j(hai , 1, k, r ): ™ k ∈ Ij \{1}, j = 1, . . . , m

n−1 qm−1 is a virtual basis of Z(U(ZG)), consisting of 2 + 2n − n − m units.

131 central units

If p is odd, then

ß i ™ ¨ i ∂ p U = ci (G, a, bp , k, 1) : 1 < k < , p k, i = 1, . . . , n 0 2 - ß S m m m pn−1 cm−j(hai , 1, k, 1)cm−j(hai , 1, k, r) ··· cm−j(hai , 1, k, r ): ™ k ∈ Ij \{1}, j = 1, . . . , m

pn−1 qm−1 is a virtual basis of Z(U(ZG)), consisting of 2 + 2pn − n − m units.

Proof We ﬁrst compute the rank of Z(U(ZG)) using the formula from Theo- rem 4.2.1. By Corollary 1.7.18, the strong Shoda pairs of G are of the following forms:

Ä ¨ pi ∂ä 1. G, Li := a, b , i = 0, . . . , n;

Ä ¨ qj ∂ä 2. hai ,Kj := a , j = 1, . . . , m. When p is odd, an easy computation shows that the rank equals

n m X Åpi−1(p − 1) ã X Åqj−1(q − 1) ã pn − 1 qm − 1 − 1 + − 1 = + − n − m, 2 2pn 2 2pn i=1 j=1 because r has odd order modulo qj, j = 1, . . . , m. When p = 2, the rank equals

n m X X Åqj−1(q − 1) ã qm − 1 2i−2 − 1 + − 1 = 2n−1 + − n − m, 2n 2n i=2 j=1

2n−1 since ab = a−1, because r has even order modulo qm. Now we use Theorem 4.4.4 to obtain a virtual basis of Z(U(ZG)):

ß pi ™ ci (G, L , k, 1) : i = 1, . . . , n, 1 < k < , p k 0 i 2 - (pn−1 ) [ Y j x c0(hai ,Kj, k, r ): j = 1, . . . , m, k ∈ Ij \{1} , x=0

132 4.5 conclusions

j where Ij is a set of coset representatives of U(Z/q Z) modulo hr, −1i containing j x 1. We claim that the units c0(hai ,Kj, k, r ), which are products of generalized m x Bass units, can be replaced by cm−j(hai , 1, k, r ), which are products of Bass units. Indeed, these units project trivially on the commutative algebra QGba. Moreover, by Proposition 4.4.3, they project to the unit

m−1 oqm (k)q ηk(ζqj )

in the simple component QGε(hai ,Kj) ' Q(ζqj ) and trivially in all other components of QG(1 − ba). By Lemma 4.4.1, the set {π ηk(ζqj ) : k ∈ Ij \{1}}

Cpn is a virtual basis of U Z[ζqj ] . This proves the claim. As we have shown, the conditions in the statement of Theorem 4.4.4 on the strong Shoda pairs of the group G are fulfilled when G is a metacyclic group of the type Cqm o1 Cpn , for two different prime numbers p and q. How- ever, the class of strongly monomial groups such that there is a complete and non-redundant set of strong Shoda pairs (H,K) of G with the property that [H : K] is a prime power, is a wider class. For example the alternating group A4 of degree 4 satisfies the condition and it is not metacyclic. Al- though, not all strongly monomial groups have only strong Shoda pairs with prime power index. It can be shown that all strong Shoda pairs of the dihe- n 2 b −1 dral group D2n = a, b : a = b = 1, a = a (respectively, the quaternion 2n 4 n 2 y −1 group Q4n = x, y : x = y = 1, x = y , x = x ) have prime power index if and only if n is a power of a prime number (respectively, n is a power of 2).

4.5 conclusions

We gave a construction for a virtual basis of Z(U(ZG)) within B1(G) in the case when G is a finite abelian-by-supersolvable group such that every cyclic subgroup of order not a divisor of 4 or 6, is subnormal in G. For finite strongly monomial groups G, we have proved that the generalized Bass units contain a subgroup of finite index in Z(U(ZG)). Moreover, when G is a strongly monomial group such that there is a complete and non-redundant set of strong Shoda pairs (H,K) of G with the property that each [H : K] is a prime power, a virtual basis within the generalized Bass units is constructed. In all results so

133 central units far, one proves that the (generalized) Bass units generate a subgroup of finite index in Z(U(ZG)). However, there are no examples known where B1(G) does not contain a subgroup of finite index in Z(U(ZG)). Therefore, one could ask and investigate whether this is true in general for all finite groups. A continuation of this work would stem from new constructions of central units for other classes of finite groups, both strongly monomial groups and non-strongly monomial. Another direction would be to generalize the known results to Z(U(RG)) for the ring of integers R of a number field.

134 5

APPLICATIONSTOUNITSOFGROUPRINGS

In this chapter, we show how the description of a complete set of orthogonal idempotents in QG together with a description of the units in Z(U(ZG)) leads to a description of the unit group U(ZG), up to finite index, for finite groups G which have no exceptional components in the Wedderburn decomposition of QG. We do this for the metacyclic groups Cqm o1 Cpn . This is merely based on the results from Theorems 1.9.2 and 1.9.3. We also show how to overcome the lack of a result as Theorem 1.9.3 in the case when QG contains an exceptional component of type EC2, but none of type EC1. If one can not construct central units, one can just allow all Bass units and use Proposition 1.13.2. Recall that finite groups G having an exceptional component in QG are classified in Corollary 3.3.1. This classification distinguishes between exceptional components of type EC1 and the ones of type EC2.

5.1 a subgroup of finite index in U (Z(Cq m o1 Cpn ))

As an application of Theorem 1.7.19 and Corollary 4.3.5, Eric Jespers, Gabriela Olteanu and Angel´ del R´ıoobtained a factorization of a subgroup of finite index of U (ZG) into a product of three nilpotent groups, and they explicitly constructed finitely many generators for each of these factors [JOdR12, Theorem 5.3]. Throughout this section p and q are different prime numbers, m and n are positive integers and G = Cq m o1 Cpn . As an application of the description of the matrix units in each simple component QGe (Corollary 2.2.5) and of the description of the central units in ZG (Corollary 4.4.5), we construct explicitly finitely many generators for three nilpotent subgroups of U (ZG) that together generate a subgroup of finite index.

135 applications to units of group rings

In order to state the next theorem, it is convenient to introduce the notation of class sum. Let G be a ﬁnite group, X a normal subgroup of G and Y a subgroup of G such that Y acts faithfully on X by conjugation. Consider the Y Y P y orbit x of an element x ∈ X, then we will call x› = y∈Y x ∈ ZX the orbit sum of x. By XgY we will denote the set of all diﬀerent orbit sums x›Y for x ∈ X.

Theorem 5.1.1

Let p and q be different prime numbers. Let G = Cqm o1 Cpn be a finite metacyclic group with Cpn = hbi and Cqm = hai. Assume that n ¨ qj ∂ either q 6= 3, or p > 2. For every j = 1, . . . , m, let Kj = a , let Fj be the center of QGε(hai ,Kj), fix a normal element wj of Q(ζqj )/Fj and let ψj : QGε(hai ,Kj) → Mpn (Fj) be the isomorphism given by The- −1 −1 −1 orem 1.5.2 with respect to wj. Let xj = ψj (P )bε(hai ,Kj)ψj (P ) , with  1 1 1 ··· 1 1   1 −1 0 ··· 0 0     1 0 −1 ··· 0 0  P =  ......  ,  ......   ......   1 0 0 · · · −1 0  1 0 0 ··· 0 −1

k n and tj be a positive integer such that tjxj ∈ ZG for all k with 1 ≤ k ≤ p . Then the following two groups are ﬁnitely generated nilpotent subgroups of U(ZG): ≠ ∑ + n 2 h −k ﬂhbi n Vj = 1 + p tj yxj bbxj : y ∈ hai , h, k ∈ {1, . . . , p }, h < k ,

≠ ∑ − n 2 h −k ﬂhbi n Vj = 1 + p tj yxj bbxj : y ∈ hai , h, k ∈ {1, . . . , p }, h > k .

+ Qm + − Qm − Hence V = j=1 Vj and V = j=1 Vj are nilpotent subgroups of U(ZG). Furthermore, the group

U, V +,V − ,

with U as in Corollary 4.4.5, is of ﬁnite index in U(ZG).

136 5.2 a method to compute U (ZG) up to commensurability

Proof It is enough to show that for each primitive central idempotent of QG, the group hU, V +,V −i contains a subgroup of finite index in U(Z(1−e)+ZGe). Since the units of the commutative components are central, we only have to Å Å ã ã flhbi consider the non-commutative components QGe ' Mpn Q hai e , with ï ò flhbi e = ε (hai ,Kj), for j = 1, . . . , m, from Corollary 1.7.18. Let O = Z hai e,

hbi hbi which is as a Z-module finitely generated by hflai e. Clearly, for y ∈ hflai , n 2 h k the elements of the form (1 − e) + (e + p tj yxj bbxj ) are in ZG and project + − trivially to QG(1 − e). By Corollary 2.2.5, the group hVj ,Vj i, generated by these elements, projects to the group

n 2 n h1 + zjEhk : zj ∈ p tj O, 1 ≤ h, k ≤ p , i 6= j, Ehk a matrix uniti

of elementary matrices of Mpn (O). If pn > 2, then the conditions of Theorem 1.9.2 are clearly satisfied. If p = 2, n = 1 and q 6= 3, the conditions of Theorem 1.9.3 are satisfied since U(O) is finite if and only if j = 1 and q = 3. This can be seen by computing the rank of U(O) for which a formula is given in Corollary 4.4.5. Hence in + − all cases hVj ,Vj i ⊆ U(ZG) is a subgroup of finite index in 1 − e + SLpn (O). By Corollary 4.4.5, U has finite index in Z(U(ZG)) and therefore it contains a subgroup of finite index in the center of 1 − e + GLpn (O). Since the center of GLpn (O) together with SLpn (O) generates a subgroup of finite index in + − GLpn (O), it follows that hU, V ,V i contains a subgroup of finite index in + the group of units of Z(1 − e) + ZGe. Now the statement follows, since Vj − and Vj correspond to upper and lower triangular matrices.

5.2 a method to compute U (ZG) up to commensurability

The proof of the following proposition is based on the Euclidean algorithm. Proposition 5.2.1 Let O be a left norm Euclidean order with norm N in either Q, a totally deﬁnite quaternion algebra over Q, or a quadratic imaginary extension of Q. If B is a Z-basis of O, then the set

ßÅ 1 x ã Å 1 0 ã ™ X = , : x ∈ B 0 1 x 1

137 applications to units of group rings

generates a subgroup of ﬁnite index in SL2 (O).

Proof We ﬁrst remark that the elementary matrices E2(O) are generated by X, because of the following matrix computations for n ∈ Z, x, y ∈ O: n Å 1 nx ã Å 1 x ã = , 0 1 0 1 Å 1 x + y ã Å 1 x ã Å 1 y ã = , 0 1 0 1 0 1

and the transposed equalities. Å a b ã Take now ∈ SL (O). We ﬁrst consider the case where both a and c d 2 c are non-zero. Since O is left norm Euclidean, there exists q1, r1 ∈ O such that a = q1c + r1 with N(r1) < N(c). An easy computation shows that

Å 1 −q ã Å a b ã Å r b − q d ã 1 = 1 1 . 0 1 c d c d

Now there exists q2, r2 ∈ O such that c = q2r1 + r2 with N(r2) < N(r1) giving

Å 1 0 ã Å r b − q d ã Å r b − q d ã 1 1 = 1 1 . −q2 1 c d r2 −q2b + q2q1d + d

Repeating this argument and since the sets {N(x) < N(a): x ∈ O} and {N(x) < N(c): x ∈ O} are ﬁnite, one can assume that there exists a Å a b ã Å ∗ ∗ ã matrix M ∈ hXi such that is either of the form M or c d 0 ∗ Å 0 ∗ ã M . ∗ ∗ Å a b ã Now assume that c = 0. We compute that ∈ SL (O) if and only 0 d 2 if N(a)N(d) = 1 and this happens if and only if a, d ∈ U(O). We can conclude that Å a b ã Å 1 bd−1 ã Å a 0 ã = . 0 d 0 1 0 d When a = 0, then b, c ∈ U(O) and

Å 0 b ã Å 1 0 ã Å 0 b ã = . c d db−1 1 c 0

138 5.2 a method to compute U (ZG) up to commensurability

Because of Dirichlet’s Theorem 1.8.8 and Kleinert’s Theorem 1.9.4, U(O) is Å a 0 ã finite. Hence, there are only finitely many matrices of the form and 0 d Å 0 b ã . This means that X generates SL (O) up to finite index. c 0 2 Due to the restrictions we obtained on the possible exceptional components in Corollary 1.9.9, together with Proposition 5.2.1, we generalize Proposi- tion 1.13.3 and Corollary 1.13.4. That is, we allow exceptional components M2(D) ' QGe of type EC2, provided one can establish a concrete isomorphism M2(D) → QGe. The proof we present is an adapted version of the proofs of [JL93, Theorem 3.3, Corollary 4.1]. For reasons of completeness, we reuse a part of their proofs. Proposition 5.2.2 Ln Ln Let G be a finite group and let QG = i=1 QGei ' i=1 Mni (Di) be the Wedderburn decomposition of QG. Assume that QG does not contain exceptional components of type EC1. Also, assume that for each integer i ∈ {1, . . . , n} such that ni 6= 1 and QGei is not exceptional (of type EC2), Gei is not fixed point free. For every exceptional component QGei ' M2(Di), Di has a left norm Euclidean order Oi. Take a Z-basis Bi of Oi and let ψi : M2(Di) → QGei be a Q-algebra isomorphism. For such i, set ß Å 0 x ã Å 0 0 ã ™ U := 1 + ψ , 1 + ψ : x ∈ B . i i 0 0 i x 0 i

S The subgroup U := hB1(G) ∪ B2(G) ∪ i Uii of QG is commensurable with U(ZG).

Proof Let i ∈ {1, . . . , n} and take a maximal order Oi in the division ring Di. Since the natural image of B1(G) in K1(ZG) is of finite index, it suffices to verify condition 2 of Proposition 1.13.2 for each simple component QGei, i.e. verify that U ∩ (1 − ei + SLni (Oi)) is of finite index in 1 − ei + SLni (Oi). If ni = 1, the result follows trivially since Di is not exceptional and hence either is commutative or a totally definite quaternion algebra. In both cases SL1(Oi) is finite (Theorem 1.9.4). If ni = 2 and M2(D) is exceptional of type EC2, then D equals Q, a quadratic imaginary extension of Q or a totally definite quaternion algebra

139 applications to units of group rings

over Q. We know by Corollary 1.9.11 that Di has a left norm Euclidean order which is the unique maximal order. Hence Oi is left norm Euclidean and it

follows from Proposition 5.2.1 that Ui ∩ (1 − ei + SLni (Oi)) is of ﬁnite index

in 1 − ei + SLni (Oi). For the remaining components QGei, we claim that there always exists a gi ∈ G such that g“iei is a non-central idempotent in QGei. Let e be a primitive central idempotent of CGei and let ρ be the irreducible representation ρ : Gei → CGe. Since Gei is not ﬁxed point free, there exists a gi ∈ G such that giei 6= ei and ρ(giei) has 1 as an eigenvalue. Diagonalizing ρ(giei), we have  1 ··· 0 0 ··· 0   ......   ......     0 ··· 1 0 ··· 0  ρ(giei) =  ,  0 ··· 0 ζj ··· 0     ......   ...... 

0 ··· 0 0 ··· ζni with 2 ≤ j ≤ ni and all ζj, . . . , ζni diﬀerent from 1. Consequently,

 1 ··· 0 0 ··· 0   ......   ......     0 ··· 1 0 ··· 0  ρ(giei) =  . “  0 ··· 0 0 ··· 0     ......   ......  0 ··· 0 0 ··· 0

It follows that ρ(g“iei) 6= 0 and ρ(g“iei) 6= 1. Hence g“iei 6= 0, g“iei 6= ei and g“iei is a non-central idempotent in QGei. Consider now the generalized bicyclic units

2 2 Bi = 1 + zi (1 − g“iei)hg“iei, 1 + zi g“ieih(1 − g“iei): h ∈ G , with zi a minimal positive integer such that zig“iei ∈ ZG. Note that, for k, l ∈ Z and x, y ∈ G,

2 k 2 l 2 (1 + zi (1 − g“iei)xg“iei) (1 + zi (1 − g“iei)yg“iei) = 1 + zi (1 − g“iei)(kx + ly)g“iei. Hence

2 2 {1 + zi (1 − g“iei)αg“iei, 1 + zi g“ieiα(1 − g“iei): α ∈ ZG} ⊆ Bi.

140 5.2 a method to compute U (ZG) up to commensurability

Since g e is a non-central idempotent in Ge ' M (D ), there exist matrix “i i Q i ni i units Ekl, 1 ≤ k, l ≤ ni such that g“iei = E11 + ... + Emm with 1 ≤ m < ni. If k ≤ m, m + 1 ≤ l ≤ ni, then

g“ieiOiEkl(1 − g“iei) = OiEkl.

Hence, as Oi is a ﬁnitely generated Z-module, there exists a positive integer nkl such that 1 + nklOiEkl ⊆ Bi. And similarly, 1 + nlkOiElk ⊆ Bi,

for some positive integer nlk. It follows that there exists a positive integer x such that 1 + xOiEkl ∈ Bi and 1 + xOiElk ∈ Bi for all 1 ≤ k ≤ m, m + 1 ≤ l ≤ ni. Now let 1 ≤ k, l ≤ m, k 6= l and α ∈ Oi. Then

2 1 + x αEkl = (1 − xαEk,m+1)(1 − xEm+1,l)(1 + xαEk,m+1)(1 + xEm+1,l) ∈ Bi.

2 Similarly, for m + 1 ≤ k, l ≤ ni and k 6= l, it follows that 1 + x OiEkl ⊆ Bi. 2 Take now Ii = x Oi. This is a non-zero ideal of Oi. Then Theorems 1.9.2

and 1.9.3 imply that 1−ei +Eni (Ii) ⊆ Bi is of ﬁnite index in 1−ei +SLni (Oi). By Lemma 1.13.1, it follows that Bi ⊆ B2(G) and hence B2(G) contains a

subgroup of finite index in 1 − ei + SLni (Oi) and the proposition follows. The set U from Proposition 5.2.2 is constructive if one can determine an explicit isomorphism ψi : M2(Di) → QGei, for each exceptional component QGei. Recall that all exceptional components of type EC2 are of the follow- √ √ √ Ä −1,−1 ä ing form: M2( ), M2( ( −1)), M2( ( −2)), M2( ( −3)), M2 , Q Q Q Q Q Ä −1,−3 ä Ä −2,−5 ä M2 , M2 ; and their maximal orders are described in Corol- Q Q lary 1.9.11. In order to describe the explicit isomorphisms ψi, one needs to construct non-central idempotents in QGei. Also, for the non-exceptional components QGej, we need non-central idempotents and therefore we ask Gej to be not fixed point free. If however, some Gej is fixed point free, but one does know another construction of non-central idempotents in QGej, then one can modify Proposition 5.2.2 and add the generalized bicyclic units based on such non-central idempotents to the set U. In this way, U still yields a subgroup commensurable with U(ZG). Some generic constructions of non-central primitive idempotents are given for nilpotent groups in Theorem 1.7.19 and for finite strongly monomial groups, with a trivial twisting for all strong Shoda pairs, in Theorem 2.2.1.

141 applications to units of group rings

5.3 examples

+ 5.3.1 U(ZD16) up to ﬁnite index

+ 8 2 5 Let G be the group with presentation D16 = a, b : a = 1, b = 1, bab = a . This is the group with SmallGroup ID [16,6]. However, this example is already well studied in [JL91, Corollary 4.10], it nicely demonstrates the use of our method. Using wedderga, we compute the Wedderburn decomposition:

QG = 4Q ⊕ 2Q(i) ⊕ M2(Q(i)).

The primitive central idempotent e associated to the last simple component 1 1 4 is afforded by the pair (hai , 1) and equals e = e(G, hai , 1) = 2 − 2 a . Hence this simple component equals the crossed product QGe = Q hai ε(hai , 1) ∗ hbi. It is easy to see that eb is a non-trivial idempotent of QGe, which affords a description of QGe as M2(bQGeb). Another simple calculation shows that Å i 0 ã (ba2eb)2 = −eb and hence the map ba2eb → defines a -algebra b b b b b 0 0 Q isomorphism between QGe and M2(Q(i)). To determine an explicit isomorphism ψ : M2(Q(i)) → QGe, it suffices to find images of the following elements:

Å1 0ã Åi 0ã Å0 1ã Å0 0ã E := ,I := ,A := ,B := . 0 0 0 0 0 0 1 0

We already know possible images for E and I:

1 1 ψ(E) = eb ∈ G, ψ(I) = bba2eb ∈ G. 4Z 4Z The images of A and B must satisfy

ψ(A) · ψ(B) = ψ(E), (20)

ψ(E) · ψ(A) · (1 − ψ(E)) = ψ(A) (21) and (1 − ψ(E)) · ψ(B) · ψ(E) = ψ(B). (22)

142 5.3 examples

Define 1 1 ψ(A) = bbae(e − b) = bbae(1 − b) ∈ G and ψ(B) = (1 − b)a−1eb ∈ G. 4Z 4Z Equations (21) and (22) are satisfied by definition and one verifies that

ψ(A)ψ(B) = eb = ψ(E).

Now we apply Proposition 5.2.2 to compute U(ZG) up to finite index. Since the components Q and Q(i) yield a finite group of units in any order, it suffices to compute the group of units of 1 − e + ZGe up to finite index, for QGe ' M2(Q(i)). Let Z[i] be the maximal order of Q(i), then U(ZGe) and SL2(Z[i]) are commensurable. From Proposition 5.2.2, we conclude that the elements

Å 0 1 ã Å 0 0 ã 1 + ψ = 1 + bae(1 − b), 1 + ψ = 1 + (1 − b)a−1eb, 0 0 b b 1 0 b b

Å 0 i ã Å 0 0 ã 1 + ψ = 1 + ba2bae(1 − b), 1 + ψ = 1 + (1 − b)a−1ba2eb 0 0 b b b i 0 b b b generate U(1 − e + ZGe) up to commensurability. Hence this set of elements in QG generates a group which is commensurable with U(ZG). 1 However these images do not lie in ZG but in 4 ZG. If one is not satisﬁed with commensurability, but if one wants to construct a list of generators within ZG, then one has to deduce from the generators of SL2(Z[i]) a list of generators of the congruence subgroup

ßÅ a + 1 b ã ™ C = : a, d ∈ 2 [i], b, c ∈ 4 [i], (a + 1)(d + 1) − bc = 1 c d + 1 Z Z

using techniques as for example Schreier’s Lemma, as shown in [JL91, Corollary 4.10]. It is easy to verify that {1 + ψ(x − 1) : x ∈ C} is contained in ZG and is of ﬁnite index in U(ZG). Schreier’s Lemma leads to lengthy and technical computations and therefore we opted to not include them in this thesis. More details can be found in [EKVG15].

5.3.2 U(ZSL(2, 5)) up to commensurability

By Theorem 1.1.2, we know that SL(2, 5) is the smallest non-solvable Frobe- nius complement (i.e. ﬁxed point free group). We apply Proposition 5.2.2 to

143 applications to units of group rings

investigate the units of ZSL(2, 5). Since none of the older techniques, such as Proposition 1.13.3 and Corollary 1.13.4, apply here, this example shows the strength of our method. Using wedderga, we compute that QSL(2, 5) is isomorphic to

Ç−1, −1å Å−1, −3ã Q ⊕ M4(Q) ⊕ √ ⊕ M2 Q( 5) Q Å−1, −1ã √ ⊕ M5(Q) ⊕ M3 ⊕ M3(Q( 5)). Q An easy GAP computation shows that for all simple components QGe diﬀer- −1,−1 ent from Q and √ , there exists a group element g such that g projects to Q( 5) b a non-central idempotent in QGe. Therefore, we can apply Proposition 5.2.2. 1 1 Consider the maximal order O = Z 1, i, (1 + j), (i + ij) in the quater- Ä ä 2 2 nion algebra −1,−3 . Let e be the primitive central idempotent associated to Q Ä −1,−3 ä Ä −1,−3 ä the component M2 and let ψ be the isomorphism between M2 Q Q and QGe. Let U be the subset of QG containing the elements Å 0 1 ã Å 0 0 ã Å 0 i ã Å 0 0 ã 1 + ψ , 1 + ψ , 1 + ψ , 1 + ψ , 0 0 1 0 0 0 i 0

Å 1 ã Å ã 0 2 (1 + j) 0 0 1 + ψ , 1 + ψ 1 , 0 0 2 (1 + j) 0 Å 1 ã Å ã 0 2 (i + ij) 0 0 1 + ψ , 1 + ψ 1 . 0 0 2 (i + ij) 0

Then the group hU ∪ B1(G) ∪ B2(G)i is commensurable with U(ZG).

5.4 conclusions

We have shown how a concrete algebra isomorphism M2(Di) → QGei for all exceptional components M2(Di) of type EC2 leads to a finite generating set (up to commensurability) of the group of units of ZG of a finite group G, provided that G does not contain exceptional components of type EC1 and all non-exceptional components are not fixed point free. Hence, to apply the technique to other finite groups than the ones presented, one should start to study the finite list of groups in Table 2 and find concrete isomorphisms of the corresponding exceptional simple component M2(D) → QGe.

144 5.4 conclusions

To extend the techniques, a first thing to do is to find constructions of non- central idempotents in the rational group algebra of fixed point free groups. Secondly, one should investigate constructions of units in orders of division rings. Recently some progress has been made in the latter topic, see [CJLdR04, JPSF09, JJK+15, BCNS]. Alternatively, the results in Chapter 2 possibly extend to group rings over number fields F . Let R be the ring of integers of F . If one can construct a generating set of Z(U(RG)), then one extends the techniques from Theorem 5.1.1 to U(RG) to generate the unit group up to finite index.

145

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154 INDEX

2-part, 22 cyclotomic unit of QG, 34 0 2 -part, 22 cyclotomic unit of Q(ζn), 34 F -critical, 72 G-code, 61 degree, 4 Q-class, 39 elementary matrix, 27 Q-conjugate, 39 equivalent valuations, 4 R-class, 39 exceptional component, 28 R-conjugate, 39 exceptional simple algebra, 28 Z-order, 24 p-adic valuation, 5 field of character values, 16 p-local index, 7 finite prime, 5 q-cyclotomic class, 54 fixed point free group, 2 action, 10 Frobenius complement, 2 archimedean valuation, 4 Frobenius group, 2

Bass unit, 34 generalized Bass unit, 111 bicyclic unit, 36 generalized bicyclic unit, 36 group algebra, 12 Cauchy sequence, 5 group code, 61 central simple algebra, 4 group ring, 12 central unit, 39 classical crossed product, 10 idempotent, 14 commensurable, 25 induced character, 17 complete, 5 inﬁnite prime, 5 complete and non-redundant set of Shoda pairs, 18 left G-code, 61 complete and non-redundant set of left group code, 61 strong Shoda pairs, 19 left norm Euclidean, 30 congruence subgroup, 26 length, 61 Congruence Subgroup Problem, 26 linear code, 61 crossed product, 10 local Hasse invariant, 6 CSP, 26 local index, 5, 7 cyclic cyclotomic algebra, 10 local Schur index, 5

155 matrix unit, 14 valuation, 4 maximal order, 24 virtual basis, 101 monomial character, 17 monomial group, 17 Z-group, 31 non-archimedean valuation, 4 normal, 3 normal basis, 3 number ﬁeld, 4 orbit sum, 136 order, 24 place, 5 prime, 5 prime lying over, 7 primitive central idempotent, 15 primitive idempotent, 14 quaternion algebra, 3 ramiﬁcation index, 7 rank, 25, 61 reduced norm, 26 residue degree, 7

Schur index, 4 semisimple, 13 Shoda pair, 17 split, 6 strong Shoda pair, 18 strongly monomial character, 18 strongly monomial group, 18 subnormal, 114 support, 12 totally deﬁnite, 3 twisting, 10 unit group, 25

156

colophon

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