The Centre of a Block

THE CENTRE OF A BLOCK

A thesis submitted to the University of Manchester for the degree of Doctor of Philosophy in the Faculty of Engineering and Physical Sciences

2016

Inga Schwabrow School of Mathematics 2 Contents

Abstract 5

Declaration 7

Acknowledgements 11

Introduction 13

1 Modular Representation Theory 17 1.1 Basic notation and setup ...... 17 1.2 Conjugacy class sums, characters and Burnside’s formula ...... 18 1.3 Blocks ...... 22 1.4 Defect and defect groups ...... 25 1.5 Jacobson radical and Loewy length ...... 28 1.6 The Reynolds ideal ...... 35

2 Equivalences of blocks 37 2.1 Brauer correspondence ...... 37 2.2 Morita equivalence ...... 38 2.3 Derived equivalence ...... 38 2.4 Stable equivalence of Morita type ...... 40 2.5 Properties of blocks with TI defect groups ...... 41 2.6 On using the centre to show no derived equivalence exists ...... 43

3 Blocks with trivial intersection defect groups 44

3 3.1 The Mathieu group M11 with p =3...... 45 3.2 The McLaughlin group McL, and Aut(McL), with p = 5 ...... 49

3.3 The Janko group J4 with p =11...... 52 3.4 The projective special unitary groups ...... 54 3.5 A question of Rickard ...... 60 3.6 On the existence of perfect isometries in blocks with TI defect groups . 62

4 On the Loewy length of the Suzuki groups and the small Ree groups in deﬁning characteristic 68 4.1 Relating structure constants ...... 68 4.2 Suzuki Groups ...... 70 4.3 The Ree groups ...... 76

5 More sporadic simple groups 108 5.1 Mathieu groups ...... 109 5.2 Janko groups ...... 111 5.3 The Higman-Sims group ...... 112 5.4 The McLaughlin group ...... 113 5.5 The Held group ...... 113 5.6 The Rudvalis group ...... 114 5.7 The O’Nan group ...... 115 5.8 A conjecture ...... 115

6 Blocks with normal, abelian defect groups 116 6.1 A reduction for the Loewy length calculation ...... 116 6.2 Calculating some examples ...... 121

n 6.3 General formulae for the Loewy length of (Cp) o E ...... 126

Bibliography 132

A GAP Code 138

B Some structure constants in Ree groups 147

Word count 21714

4 The University of Manchester

Inga Schwabrow Doctor of Philosophy The Centre of a Block June 6, 2016 Let G be a finite group and F a field. The group algebra FG decomposes as a direct sum of two-sided ideals, called the blocks of FG. In this thesis the structure of the centre of a block for various groups is investigated. Studying these subalgebras yields information about the relationship between two block algebras and, in certain cases, forms a vital tool in establishing the non-existence of an important equivalence in the context of modular representation theory. In particular, the focus lies on determining the Loewy structure for the centre of a block, which so far has not been studied in detail but is fundamental in gaining a better understanding of the block itself. For finite groups G with non-abelian, trivial intersection Sylow p-subgroups, the analysis of the Loewy structure of the centre of a block allows us to deduce that a stable equivalence of Morita type does not induce an algebra isomorphism between the centre of the principal block of G and its Sylow normaliser. This was already known for the Suzuki groups; the techniques will be generalised to extend the result to cover 2 the Ree groups of type G2(q). In addition, the three sporadic simple groups with the trivial intersection property, M11, McL and J4, together with some small projective special unitary groups are studied with respect to showing the non-existence of an isomorphism between the centre of the principal block and the centre of its Brauer correspondent. Finally, the Loewy structure of centres of various principal blocks are calculated. In particular, some small sporadic simple groups and groups with normal, elementary abelian Sylow p-subgroups are considered. For the latter, some specific formulae for the Loewy length are derived, which generalises recent results on groups with cyclic Sylow p-subgroups.

5 6 Declaration

No portion of the work referred to in the thesis has been submitted in support of an application for another degree or qualiﬁcation of this or any other university or other institute of learning.

7 8 Copyright Statement

i. The author of this thesis (including any appendices and/or schedules to this thesis) owns certain copyright or related rights in it (the “Copyright”) and she has given The University of Manchester certain rights to use such Copyright, including for administrative purposes.

ii. Copies of this thesis, either in full or in extracts and whether in hard or electronic copy, may be made only in accordance with the Copyright, Designs and Patents Act 1988 (as amended) and regulations issued under it or, where appropriate, in accordance with licensing agreements which the University has from time to time. This page must form part of any such copies made. iii. The ownership of certain Copyright, patents, designs, trade marks and other intellectual property (the “Intellectual Property”) and any reproductions of copyright works in the thesis, for example graphs and tables (“Reproductions”), which may be described in this thesis, may not be owned by the author and may be owned by third parties. Such Intellectual Property and Reproductions cannot and must not be made available for use without the prior written permission of the owner(s) of the relevant Intellectual Property and/or Reproductions. iv. Further information on the conditions under which disclosure, publication and com- mercialisation of this thesis, the Copyright and any Intellectual Property and/or Reproductions described in it may take place is available in the University IP Policy (see http://documents.manchester.ac.uk/DocuInfo.aspx?DocID=487), in any relevant Thesis restriction declarations deposited in the University Library, The Univer- sity Library’s regulations (see http://www.manchester.ac.uk/library/aboutus/regulations) and in The University’s Policy on Presentation of Theses.

9 10 Acknowledgements

First of all, I would like to thank my supervisor, Dr. Charles Eaton, for his support, guidance and assistance given to me throughout my PhD. Without the many meetings and helpful discussions, none of this research would have been possible. I would also like to thank my Master supervisor at the University of Southampton, Dr. Tim Burness, for introducing me to the representation theory of finite groups, an extremely rich and beautiful area of pure mathematics. Additional thanks go to Professor Radha Kessar for making me aware of the paper discussed in Section 3.4.11 and Dr. Benjamin Sambale for some useful discussions on Loewy lengths including the use of GAP in the calculations. I greatly appreciate the Engineering and Physical Sciences Research Council for funding my PhD research. Many thanks go to all staff and fellow students at the School of Mathematics, University of Manchester, who have helped me in any way over the last three and a half years. Finally, I thank my family for their continuous support throughout my studies and especially Julian Brough, for his patience and support, which helped me overcome many tough situations and finish this thesis.

11 12 Introduction

The representation theory of finite groups has many fascinating and deep open prob- lems. Many of them originate in the work of Richard Brauer, who began the systematic study of modular representations of a finite group, in which the characteristic of the field divides the order of the group. His conjectures formulated in 1963 [Bra63] have led to the development of many new concepts and methods to investigate properties of finite groups. This work often concerns local representation theory, which asserts that fundamental information about the representation theory of G is encoded in some local subgroup of G, that is, the normaliser of some non-trivial p-subgroup. Let p be a fixed prime number. Let G be a finite group whose order is divisible by p. Let O be a complete valuation ring whose residue field k has characteristic p. The group algebra OG decomposes into blocks; hence there is an inherent desire to study these blocks up to some notion of equivalence. Different levels of equivalence between two block algebras A and B have been defined and studied; we are most interested in the notion of a derived equivalence, which has the following consequences:

Derived equivalence ⇒ Perfect Isometry ⇒ Z(A) ∼= Z(B).

This motivates us to study the centre of a block. So far, this algebra is not well understood and so we explicitly calculate its structure for various groups. A group is said to have trivial intersection (TI) Sylow p-subgroups groups, if any two distinct Sylow p-subgroups intersect trivially. If a block B of G has TI defect groups P , then there exists a stable equivalence of Morita type between B and its

Brauer correspondent b in NG(P ). This equivalence induces an algebra homomorphism between the respective stable centres, where Zst(A) := Z(A)/Zpr(A). Although Z(B) and Z(b) have the same dimension [BM90, Theorem 9.2], and the stable centres are isomorphic, i.e. Zst(B) ∼= Zst(b) [Bro94, Proposition 5.4], surprisingly, there is in

13 14 general no isomorphism between the two centres of B and b.

The abelian defect group conjecture proposed by Brou´ein 1990 [Bro90, Question 6.2] provides a structural explanation for the close relationship between certain blocks. The conjecture claims that there is a derived equivalence between a block B with abelian defect groups and its Brauer correspondent b in NG(P ) where P ∈ Sylp(G). An important consequence of Brou´e’sconjecture is that if a block has abelian defect groups then the derived equivalence induces an isomorphism between the centres of the blocks B and b [Bro90, Theorem 1.5].

In this thesis we will show the non-existence of such an isomorphism under the diﬀerent assumption that the defect groups have the trivial intersection property, and

1+2 are not abelian or of the form p− . The Loewy length of an algebra is deﬁned to be the nilpotency length of its Jacob- son radical. Calculating the Loewy length of the centre of a block forms an important tool in establishing the non-existence of an isomorphism between two centres of blocks. Most notably, we prove that in characteristic 3, the centre of the principal block of the

2 Ree group, G2(q), has Loewy lengths 3 while the centre of its Brauer correspondent has Loewy length 2. It therefore follows that there is no perfect isometry and hence no derived equivalence between the two blocks. In other groups, the diﬀerence in the dimension of the radical squared allows us to draw the same conclusion.

In the last part of this thesis, we concentrate on calculating the Loewy length of centres of some group algebras with normal, elementary abelian Sylow p-subgroups. As mentioned before, not much is known about the structure of centres of blocks, and in particular our work generalises recent results on the Loewy length of centres of blocks with cyclic defect groups.

In Chapter 1 we recall some standard deﬁnitions, notation and results relating to modular representation theory. Most of our calculations rely heavily upon Burnside’s formula. Therefore, Section 1.2 is dedicated to outlining the details and basic properties of this formula. We then introduce the fundamental notion of a block and the main concepts related to blocks, such as defect and defect groups. Section 1.5 is dedicated to the deﬁnition and properties of the Jacobson radical of an algebra, and the associated Loewy length. While the Loewy length of a block has been studied, not 15 as many results are known concerning the Loewy length of the centre of a block. We summarise some known results and calculate some explicit examples.

The aim of Chapter 2 is to give a brief introduction to the diﬀerent levels of equivalence which can exist between blocks and to highlight some key consequences which are important for our work. Brou´e’sabelian defect group conjecture is of most importance to us as it motivates our problem; the required details of this conjecture will be described in Section 2.3.1. Instead of abelian defect groups, we consider the notion of trivial intersection defect groups; Brauer corresponding blocks with this property are stably equivalent of Morita type via the Green correspondence. Some details relating to this property will be discussed in Section 2.5. The last section provides a basic outline of the approach we will use to establish the non-existence of a perfect isometry in the trivial intersection case.

In Chapter 3, we study the principal block of some groups with trivial intersection Sylow p-subgroups. In particular, we concentrate on the three sporadic simple groups with this property: M11, McL and J4 with the primes 3, 5 and 11 respectively. Fur- thermore, some small projective special linear groups with TI Sylow p-subgroups are considered. One of the properties of blocks with TI defect groups is that the centres of the two Brauer corresponding blocks have the same dimension. However, the main result in this chapter shows that for these examples, if the Sylow p-subgroups are not abelian, then there is no isomorphism between the two centres. In Section 3.5, we brieﬂy discuss how the main result relates to a question proposed by Rickard, which asks if tensor products respect the property of stable equivalence of Morita type. In particular, it is explained how blocks with TI defect groups provide counterexamples to this question. The overall results presented in this chapter will be linked together in Theorems 3.6.8 and 3.6.9, which form two of the fundamental theorems of this thesis.

The starting point for the research in this thesis was a paper by Cliﬀ [Cli00], which provided the ﬁrst example of two blocks with TI defect groups which are not derived equivalent over O. We give an outline of this result at the start of Chapter 4. The

2 main focus of the rest of this chapter is on the Ree groups G2(q). The main result is 2 Theorem 4.3.23 which states that for G = G2(q), due to diﬀerent Loewy lengths, the centre of the principal 3-block and the centre of its Brauer correspondent in NG(P ) are not isomorphic over k. As part of this work, we also establish the full character 16 table of the normaliser of a Sylow 3-subgroup, given in Table 4.3.7. Chapter 5 consists of an overview of the Loewy lengths of the centre of the principal block of a sporadic simple group and its Brauer correspondent in NG(P ). The calculations were carried out in GAP and the code can be found in Appendix A. In Chapter 6 we calculate the Loewy length of some groups with normal, elementary abelian p-subgroups. The ﬁrst part presents a reduction for the calculation of the Loewy length which is fundamental in the proofs of general formulae given in Section 6.3. Work done in gathering evidence to support our initial observations is provided in Section 6.2; here we explicitly calculate various Loewy lengths using GAP. Chapter 1

Modular Representation Theory

This chapter covers the background material for this thesis and introduces the notation which is used throughout. For a standard reference on modular representation theory, the reader is referred to one of the following textbooks: [Alp86], [Ben95], [CR81] or [Nav98].

1.1 Basic notation and setup

Throughout this thesis, G denotes a ﬁnite group, p a prime number, k a ﬁeld of characteristic p, and kG denotes the group algebra of G over k. For g, h ∈ G, conjugation of g by h is given by gh := h−1gh and the conjugacy class

G −1 of an element g in G is denoted by CG(g) = g := {h gh | h ∈ G}; when the group

G is clear from the context, the subscript G is dropped, i.e. C (g) = CG(g). Let cc(G) denote the set of conjugacy classes inside G, and P denotes a set of representatives of the conjugacy classes of G. For two subgroups Q, P of G we use Q ≤G P to denote that there is an element g ∈ G with g−1Qg ≤ P . For H a subgroup of G, denoted H ≤ G, the centraliser of H in G is given by

−1 CG(H) = {g ∈ G | g hg = h ∀h ∈ H} and the normaliser is given by NG(H) = −1 {g ∈ G | g Hg = H}. The centre of G is denoted by Z(G). If p is a prime, Op(G) denotes the largest normal p-subgroup of G. Similarly, Op0 (G) is the largest normal p0-subgroup of G. We use the standard notation |G| to denote the order of a group G, and for a prime p, |G|p denotes the maximal power of p dividing |G|. An element g ∈ G is called

17 18 CHAPTER 1. MODULAR REPRESENTATION THEORY p-singular if p divides the order of g, while if p does not divide the order of g, the element is called p-regular. If C (g) is a conjugacy class of G, we denote the set of all G-conjugates of any Sylow p-subgroup of CG(g) by δ(C (g)); this set is independent of the chosen representative g ∈ C (g). Each element of δ(C (g)) is called a defect group of C (g), and their common

dg order is given by p with dg called the defect of the conjugacy class C (g). At times, the notation used will be taken from the Atlas [CCN+85]; in particular conjugacy classes are denoted by C (n−) such that elements in this class have order n and “ − ” is replaced by a capital letter to denote the chosen conjugacy class. Moreover, for groups A and B, A : B denotes a split extension of A by B and A ∗ B

1+2n denotes a central product of A by B; p+ denotes an extraspecial group of order 1+2n 1+2n 1+2n p and exponent p, and p− denotes an extraspecial group of order p and exponent p2.

1.2 Conjugacy class sums, characters and Burn- side’s formula

In this section, F denotes an arbitrary ﬁeld. Let Cb(g) denote the class sum of the P conjugacy class C (g), that is Cb(g) = h∈C (g) h ∈ FG. The set of conjugacy class sums forms a basis for the centre of the group algebra FG.

Lemma 1.2.1. Let C (y1),..., C (yr) be the conjugacy classes of G. Then the set

{Cb(yi) | 1 ≤ i ≤ r} forms a basis for Z(FG). Proof. For any g ∈ G we have

−1 X −1 X g Cb(yi)g = g xg = z = Cb(yi),

x∈C (yi) z∈C (yi) thus Cb(yi) ∈ Z(FG). Moreover, since the conjugacy classes are pairwise disjoint, the conjugacy class sums are linearly independent. P Let h∈G αhh ∈ Z(FG) where αh ∈ F. For any g ∈ G,

X −1 −1 X X αh(g hg) = g ( αhh)g = αhh. h∈G h∈G h∈G

Hence if h ∈ G then αg−1hg = αh, i.e. elements in the same conjugacy class have the same coeﬃcient. In particular, the set of conjugacy class sums spans Z(FG). 1.2. CONJUGACY CLASS SUMS, CHARACTERS , BURNSIDE’S FORMULA 19

1.2.1 Characters

In this section we briefly introduce a couple of results relating to the ordinary irreducible characters of a group; for more background on these, the reader is referred to [JL01]. The set of irreducible characters of G over the field F is defined to be IrrF(G) := {χ | χ is an irreducible character of G over F}. Furthermore, we set

Irr(G) := IrrC(G). We state the well known orthogonality relations between the rows and the columns of a character table over C.

Theorem 1.2.2. [JL01, Theorem 16.4]

(i) Let α, β ∈ Irr(G). Then  1 X  1, if α = β; hα, βi := α(g)β(g) = |G| g∈G  0, if α 6= β.

(ii) Let g, h ∈ G. Then  X  |CG(g)|, if g is conjugate to h; χ(g)χ(h) = χ∈ Irr(G)  0, otherwise.

Let φ be a class function on H ≤ G, i.e. a function which is constant on conjugacy classes. Extend φ to a function φ◦ on G by setting φ◦(g) = 0 for g 6∈ H. We deﬁne G the induced class function IndH (φ): G → C by 1 X IndG φ(g) := φ◦(x−1gx) H |H| x∈G for all g ∈ G (see [JL01, Proposition 21.19]). Let H be a normal subgroup of G; in G this case, IndH χ(g) = 0 for any g 6∈ H.

Lemma 1.2.3. [JL01, Proposition 17.14] Suppose χ ∈ Irr(G) is irreducible and λ ∈

Irr(G) is linear, i.e. λ(1G) = 1. Then λ · χ, deﬁned by λ · χ(g) = λ(g) · χ(g), is also irreducible.

1.2.2 Burnside’s formula

As the conjugacy class sums form a basis of Z(FG) (by Lemma 1.2.1), the multiplication of any two such sums must be a sum of conjugacy class sums. Therefore the 20 CHAPTER 1. MODULAR REPRESENTATION THEORY following notation is adopted: X Cb(x)Cb(y) = a(x, y, z)Cb(z) for x, y ∈ G, (1.2.1) z∈P where P is as deﬁned in Section 1.1, i.e. a set of representatives of the conjugacy classes of G. The non-negative integers a(x, y, z) are referred to as the class algebra constants. By deﬁnition, the constants a(x, y, z) are given by

a(x, y, z) := |{(g, h) | g ∈ C (x), h ∈ C (y), gh = z}| ∈ N0, which is independent of the choice of z ∈ C (z). If z = 1G then   0, if y 6= x−1; a(x, y, z) =  |C (x)|, if y = x−1.

Burnside’s original work in representation theory over C provided a method for ob- taining the class algebra constants from the character table of the group. In particular, this connection is made precise by Burnside’s formula [Bur11, p.316] which forms a crucial part in the study of representation theory and will play a large role in the calculations to follow in this thesis. Given x, y, z ∈ G, Burnside’s formula states: |G| X χ(x)χ(y)χ(z−1) a(x, y, z) = ; (1.2.2) |CG(x)||CG(y)| χ(1) χ∈Irr(G) for a proof see for example [Kar92, Theorem 1.2. p.798]. The following two results can be found in [CR62, Lemma 87.4, Cor 87.7].

Lemma 1.2.4. Given x, y, z ∈ G, then |C (z)| a(x, y, z) = G a(x−1, z, y). |CG(y)|

Proof. Since a(x, y, z) ∈ N0 we have |G| X χ(x−1)χ(z)χ(y−1) a(x−1, z, y) = |CG(x)| |CG(z)| χ(1) χ∈Irr(G) |G| X χ(x−1)χ(z)χ(y−1) = |CG(x)| |CG(z)| χ(1) χ∈Irr(G) |G| X χ(x)χ(y)χ(z−1) = |CG(x)| |CG(z)| χ(1) χ∈Irr(G) = |CG(y)| a(x, y, z) |CG(z)| 1.2. CONJUGACY CLASS SUMS, CHARACTERS , BURNSIDE’S FORMULA 21

Theorem 1.2.5. Let k be a ﬁeld of characteristic p, and for g ∈ G, let dg be the defect of the conjugacy class C (g). If dy < dz then p | a(x, y, z). Therefore X Cb(x)Cb(y) = a(x, y, z)Cb(z) ∈ kG.

dz≤dy Proof. The theorem follows directly from Lemma 1.2.4.

Lemma 1.2.6. Given x, y, z ∈ G, then a(x, y, z) = a(x−1, y−1, z−1).

−1 −1 Proof. Observe that for g ∈ G, |CG(g)| = |CG(g )| and χ(g) = χ(g ). Hence |G| X χ(x)χ(y)χ(z−1) a(x, y, z) = |CG(x)| |CG(y)| χ(1) χ∈Irr(G) |G| X χ(x−1)χ(y−1)χ(z) = −1 −1 |CG(x )| |CG(y )| χ(1) χ∈Irr(G) |G| X χ(x−1) χ(y−1) χ(z) = −1 −1 |CG(x )| |CG(y )| χ(1) χ∈Irr(G) |G| X χ(x−1)χ(y−1)χ(z) = −1 −1 |CG(x )| |CG(y )| χ(1) χ∈Irr(G) = a(x−1, y−1, z−1) = a(x−1, y−1, z−1)

−1 −1 −1 The ﬁnal equality is due to the fact that a(x , y , z ) ∈ N0.

Lemma 1.2.6 reduces the number of explicit calculations required. In particular, for x, y ∈ G, the structure constants arising in the multiplication of Cb(x)Cb(y) yield the multiplication of Cb(x−1)Cb(y−1).

Proposition 1.2.7. Fix y, z ∈ G. Then

X a(x, y, z) = |C (y)|. x∈P Proof. We have

X X X X X Cb(x)Cb(y) = gCb(y) = Gyˆ 0 = |C (y)| g = |C (y)| Cb(z). x∈P g∈G y0∈C (y) g∈G z∈P On the other hand

X X X Cb(x)Cb(y) = a(x, y, z)Cb(z). x∈P x∈P z∈P P Therefore the coeﬃcient of Cb(z) is x∈P a(x, y, z) and the proposition follows. 22 CHAPTER 1. MODULAR REPRESENTATION THEORY

1.2.3 Extension of Burnside’s formula

The formula for the product of two conjugacy class sums given in Equation 1.2.2 is a classical result; however it can be generalised to compute the product of several conjugacy class sums from the character table.

Let P = {yj|1 ≤ j ≤ m} be a full set of representatives for the conjugacy classes of the ﬁnite group G, and for 1 ≤ j ≤ m, let Cb(yj) = Cbj. Let {χi|1 ≤ i ≤ m} be the set of complex irreducible characters of G.

Lemma 1.2.8. [Rob84] Let n1, n2, . . . , nm be non-negative integers. Then for any

Qm nj x ∈ G the coeﬃcient of x in the product j=1(Cbj) is given by

Pm m m ( nj )−1 nj |G| j=1 X Y χi(yj) m χi(1)χi(x) Q |C (y )|nj χ (1) j=1 G j i=1 j=1 i

In particular, for consistency of notation, if x, y, z ∈ G, then the product of the three corresponding conjugacy class sums is written as

X Cb(x)Cb(y)Cb(z) = a(x, y, z, w)Cb(w) (1.2.3) w∈P where

|G|2 X χ(x)χ(y)χ(z)χ(w−1) a(x, y, z, w) = 2 (1.2.4) |CG(x)| |CG(y)| |CG(z)| [χ(1)] χ∈Irr(G)

Note that when one of x, y or z is trivial, we obtain the structure constant for the multiplication of two conjugacy class sums as given in Equation 1.2.2.

1.3 Blocks

This section introduces the fundamental concept of a block. There are various equivalent ways to deﬁne blocks, for example through idempotents, Brauer characters or central characters. In this section we will introduce the notion of blocks via idempotents. Any algebra can be decomposed into a direct sum of indecomposable summands, which can then be studied individually. The decomposition of an algebra A is related to the existence of idempotents in A. 1.3. BLOCKS 23

Deﬁnition 1.3.1. Let A be an algebra over a ﬁeld F. A non-zero element e ∈ A is called an idempotent if e2 = e; if moreover e ∈ Z(A), then the element is called a central idempotent. Two idempotents e1 and e2 in A are called orthogonal if e1e2 = e2e1 = 0. An idempotent e is called primitive if it cannot be written as the sum of two orthogonal idempotents.

Before we discuss blocks, we fix the fields which will be considered throughout this thesis. In particular, as representations over fields of characteristic zero are well understood, we consider fields which arise as a quotient field of such characteristic zero fields.

Definition 1.3.2. A p-modular system is a triple (K, O, k) where K is a field of characteristic zero, O is a complete valuation ring with unique maximal ideal J(O), and k is a field of characteristic p; in addition O/J(O) ∼= k and K is the field of fractions of O.

A ﬁeld K is called suﬃciently large relative to G if it contains all the mth roots of

m unity, where m is the exponent of G (i.e. m is the smallest integer such that g = 1G for all g ∈ G).

Remark. Throughout this thesis we will be working with a finite group G. Therefore p-modular systems such that the characteristic of k divides the order of G and the system is sufficiently large for G are considered. If we are working with two finite groups G and H at the same time, then we will assume that (K, O, k) is sufficiently large for both groups. Furthermore, we only consider the cases where k is algebraically closed.

We now restrict this concept to the context of group algebras; for further details see [Ben95, Section 6.1]. Let kG be a group algebra and

kG = B1 ⊕ B2 ⊕ ... ⊕ Br be the decomposition of kG into a direct sum of indecomposable two-sided ideals. This

0 0 0 is unique in the sense that if kG = B1 ⊕ B2 ⊕ ... ⊕ Bs is another such decomposition, 0 then r = s and up to reordering Bi = Bi for all i, 1 ≤ i ≤ r. We call Bi the blocks of kG, and Bl(G) denotes the set of blocks of kG. 24 CHAPTER 1. MODULAR REPRESENTATION THEORY

This decomposition corresponds to a decomposition of the identity element

1 = e1 ⊕ e2 ⊕ ... ⊕ er as a sum of orthogonal, primitive, central idempotents; the correspondence is given by

Bi = kGei and we call ei the block idempotent of Bi. If (K, O, k) is a p-modular system, as both Z(OG) and Z(kG) have a basis consisting of the conjugacy class sums in G, it follows that reduction modulo J(O) is a surjective homomorphism Z(OG) Z(kG). Hence the idempotents ei ∈ kG may be lifted to orthogonal primitive central idempotentse ˆi ∈ OG. We thus have ˆ ˆ ˆ OG = B1 ⊕ B2 ⊕ ... ⊕ Br

= OGeˆ1 ⊕ OGeˆ2 ⊕ ... ⊕ OGeˆr l l l

kG = kGe1 ⊕ kGe2 ⊕ ... ⊕ kGer

= B1 ⊕ B2 ⊕ ... ⊕ Br

If M is a kG-module, it follows that M = Me1 ⊕ Me2 ⊕ ... ⊕ Mer. Therefore if M is indecomposable, M = Mei for a unique i and Mej = 0 for j 6= i; thus we say that

M belongs to the unique block Bi. As K is of characteristic zero, any indecomposable module is irreducible and thus if M is an irreducible KG-module, then Meˆi = M for a unique i, and Meˆj = 0 for j 6= i.

Deﬁnition 1.3.3. The block of G containing the trivial module is called the principal p-block of G, denoted by B0(G) = B0(kG) = kGe0.

The primitive central idempotente ˆχ corresponding to the irreducible character

χ ∈ IrrK (G) is given by χ(1) X eˆ := χ(g−1)g ∈ Z(KG). (1.3.1) χ |G| g∈G ˆ Fixing a block Bi, the block idempotente ˆi is given by   X 1 X X eˆ = e = χ(1)χ(g−1) g ∈ OG. (1.3.2) i χ |G|   χ∈Irr(Bi) g∈G χ∈Irr(Bi)

P Theorem 1.3.4. [K¨ul81,Lemma K] Let e = g∈G αgg be the block idempotent of B.

Then αg = 0 for all p-singular elements g ∈ G. 1.4. DEFECT AND DEFECT GROUPS 25 1.4 Defect and defect groups

One can associate to each block B of G a conjugacy class of p-subgroups of G, called the defect groups of the block. In some sense, these groups determine how complicated ∼ the block is; in particular {1} is a defect group of B if and only if B = Matn(k). There are many equivalent ways to characterise defect groups. One being that they are the minimal subgroups D of G such that every kG-module belonging to the block is a direct summand of a module induced from D. We start by deﬁning the Brauer map, an algebra homomorphism between two related centres of group algebras [Nav98, Theorem 4.9, p.85].

Deﬁnition 1.4.1. Suppose that P is a p-subgroup of G and let CG(P ) ⊆ H ⊆ NG(P ).

We deﬁne an algebra homomorphism BrP : Z(kG) → Z(kH) by deﬁning BrP on Cb(x) for x ∈ P and extending linearly:

X 0 BrP (Cb(x)) = x . 0 x ∈C (x)∩CG(P ) Theorem 1.4.2. [Nav98, Theorem 4.10, p.86]. If P is any p-subgroup of G then the kernel of the Brauer homomorphism is spanned by the conjugacy class sums such that P is not contained in any G-conjugate of a defect group of the conjugacy class. In other words, X I := ker(BrP ) = kCb(x)

P *GDx where Dx ∈ δ(C (x)).

Definition 1.4.3. [Kül91]For any p-subgroup P of G, define IP (kG) := Z(kG) ∩ T Q ker(BrQ), where Q ranges over all p-subgroups of G satisfying Q 6≤G P ; it is an ideal of Z(kG). This is equivalent to (see [Kül91, p. 439])

IP (kG) = spank{Cb(x) | C (x) ∈ cc(G),Dx ≤G P for Dx ∈ δ(C (x))}.

In particular, we deﬁne I1(kG) to be the subspace of Z(kG) spanned by all class sums of defect 0, i.e.

I1(kG) = spank{Cb(x) | C (x) ∈ cc(G),Dx = {1} for Dx ∈ δ(C (x))}.

If P and Q are conjugate p-subgroups of G, then IP (kG) = IQ(kG). Furthermore, if

P is a Sylow p-subgroup of G then IP (kG) = Z(kG). 26 CHAPTER 1. MODULAR REPRESENTATION THEORY

Deﬁnition 1.4.4. Let B be a p-block of G with block idempotent e.A p-subgroup

D of G is called a defect group for B if e ∈ ID(kG) but e 6∈ IQ(kG) for any proper subgroup Q of D. We will denote the set of defect groups of B by δ(B).

We now give the deﬁnitions of the defect of a character and of a block.

Deﬁnition 1.4.5. For χ ∈ Irr(G) we deﬁne the defect, d(χ), of χ to be the exact |G| power of p dividing the integer χ(1) , i.e.

d(χ) p = |G|p − χ(1)p.

We then deﬁne the defect d(B) of a block B to be the maximal defect of an irreducible character in B, i.e.

d(B) := maxχ∈Irr(B){d(χ)}.

Furthermore, if χ ∈ Irr(B) then

ht(χ) := d(B) − d(χ) ≥ 0 is called the height of the character χ.

Note that by deﬁnition, every block contains at least one irreducible character of height zero. If D is a defect group of a block of kG then |D| = pd(B) [Nav98, Theorem 4.6]. Further, if D is also a Sylow p-subgroup of G, then we say that B has maximal defect. The principal block always has maximal defect; this follows directly from the fact that it contains the trivial character.

Lemma 1.4.6 (Brauer-Nesbitt). [BN41] Let G be a ﬁnite group and D a Sylow p- subgroup of G. Then the number of blocks of kG with defect group D is equal to the number of p-regular conjugacy classes with defect group D.

Irreducible characters of defect 0 are somewhat special; each such character forms a block of their own.

Theorem 1.4.7. [LP10, Theorem 4.4.14] Let B be a block of kG and let χ ∈ Irr(B). Then the following are equivalent:

1. Irr(B) = {χ};

2. d(B) = 0; 1.4. DEFECT AND DEFECT GROUPS 27

3. d(χ) = 0;

4. χ(g) = 0 for all p-singular elements g ∈ G;

5. χ(1)p = |G|p.

Defect groups of blocks have a number of interesting properties. Every defect group D of a block B of G is a Sylow p-subgroup of the centraliser of a p-regular element in G [Bra56, 10C]. Green was ﬁrst to observe that a defect group is also always an intersection of at most two Sylow p-subgroups.

Theorem 1.4.8 (Green). [Gre68, Theorem 3] Let B be a p-block of G with defect group

g D. Let P ∈ Sylp(G) with D ⊆ P . Then D = P ∩ P for some element g ∈ CG(D).

Since Op(G) is the intersection of all Sylow p-subgroups of G, we have Op(G) ≤ D for every D ∈ δ(B).

Deﬁnition 1.4.9. We say that D is a trivial intersection (TI) subgroup of G if for

g each g ∈ G \ NG(D) we have D ∩ D = {1}.

Remark 1.4.10. An important consequence of Green’s theorem, Theorem 1.4.8, is that for groups with trivial intersection Sylow p-subgroups, if the group algebra is decomposable, then it splits into blocks of full defect (where the Sylow p-subgroups are the defect groups) and blocks of defect zero (the defect groups must be trivial).

Corollary 1.4.11. If D is a defect group of a p-block of G, then D = Op(NG(D)).

Proof. It is clear that D ⊆ Op(NG(D)), since D is a normal p-subgroup of NG(D).

For the opposite inclusion, suppose P ∈ Sylp(G) such that S ≤ P where S ∈

Sylp(NG(D)). Then P ∩ NG(D) ∈ Sylp(NG(D)). g g By Theorem 1.4.8, D = P ∩P for some g ∈ CG(D) ≤ NG(D). Thus P ∩NG(D) = g g (P ∩ NG(D)) ∈ Sylp(NG(D)). Hence D = (P ∩ NG(D)) ∩ (P ∩ NG(D)) is the intersection of two Sylow p-subgroups of NG(D), and therefore Op(NG(D)) ⊆ D.

Deﬁnition 1.4.12. Let N ¢ G and b ∈ Bl(N). Then B ∈ Bl(G) is said to cover b if eBeb 6= 0 where eB ∈ Z(kG) and eb ∈ Z(kN) are the block idempotents of B and b respectively. The set of blocks of G covering b ∈ Bl(B) will be denoted by Bl(G | b). 28 CHAPTER 1. MODULAR REPRESENTATION THEORY

Lemma 1.4.13. [Gor80, Theorem 6.3.2] If G is p-solvable with Op0 (G) = 1, then

CG(Op(G)) ≤ Op(G).

Lemma 1.4.14. [LP10, Corollary 4.7.29] Assume that CG(Op(G)) ≤ Op(G). Then |Bl(G)| = 1.

Proof. Since Op(G) is a p-group, Op(G) has only one block B0(Op(G)). Hence every block B ∈ Bl(G) covers B0(Op(G)) (since eb = 1 so eBeb = eB 6= 0.)

If D ∈ δ(B) then Op(G) ≤ D by Remark 1.4.10. Hence CG(D) ≤ CG(Op(G)) ≤ G Op(G), so B is the only p-block of G covering b and B = B0(Op(G)) (see [LP10, Lemma 4.7.28(e)]).

In chapter 6, p0-extensions of p-groups are considered. In particular, these groups are p-solvable and so the following result will be of use to the theory.

Proposition 1.4.15. [Kar87, Proposition III.1.12] Let G be a p-solvable group. Then kG is indecomposable if and only if Op0 (G) = 1.

1.5 Jacobson radical and Loewy length

For any ring A, the Jacobson radical of A, denoted J(A), is deﬁned by

J(A) = {a ∈ A | aM = 0 for all simple A-modules M}.

This is a two-sided ideal and hence a submodule of A. In fact J(A) is the largest nilpotent ideal of A, in the following sense:

Deﬁnition 1.5.1. An ideal I of A is called nilpotent if In = 0 for some n ∈ N. Here n I consists of all elements of the form x1x2 . . . xn where x1, x2, . . . , xn ∈ I.

We have the following theorem (for the proof see [Alp86, p.3 Theorem 3]).

Theorem 1.5.2. The Jacobson radical of A is equal to each of the following:

1. the smallest submodule of A whose corresponding quotient is semisimple;

2. the intersection of all the maximal submodules of A;

3. the largest nilpotent ideal of A. 1.5. JACOBSON RADICAL AND LOEWY LENGTH 29

A numerical invariant which we want to calculate is the Loewy length of the centre of a block B.

Deﬁnition 1.5.3. Let A be an algebra. The Loewy length of A, denoted LL(A), is deﬁned to be the least positive integer n such that J n(A) = 0 but J n−1(A) 6= 0, where J(A) denotes the Jacobson radical of A. In the literature, this integer is often also referred to as the nilpotency index of J(A), denoted t(A).

We now list a couple of important equalities. Let A be a ﬁnite dimensional algebra over a ﬁeld k. If A decomposes into n subalgebras, A = A1 ⊕ ... ⊕ An, then

Z(A) = Z(A1) ⊕ ... ⊕ Z(An),

J(A) = J(A1) ⊕ ... ⊕ J(An), and J(Z(A)) = J(A) ∩ Z(A) [NT89, Theorem 3.10]. By Lemma 1.2.1, the conjugacy class sums form a basis for Z(kG); hence dimk(Z(kG)) = |cc(G)|.

Theorem 1.5.4. [Kar87, Proposition I.20.9] Let n be the number of blocks of kG, i.e. n = |Bl(G)|. Then dimk(J(Z(kG))) = dimk(Z(kG)) − n.

Remark. For a block B, the algebra Z(B) is a local algebra [Nav98, Corollary 3.12], so it has a unique maximal ideal. Since the Jacobson radical is the intersection of all maximal ideals, the Jacobson radical of the block is simply the unique maximal ideal. Hence dim(J(Z(B))) = dim(Z(B)) − 1.

1.5.1 Jacobson radical of the group algebra

The results presented in this section are summarised in [Kar87]; the original proofs are due to various authors.

Proposition 1.5.5. [Kar87, Proposition III.1.6] Let H ≤ G. Then

1. J(kG) ∩ kH ⊆ J(kH);

2. dimk(J(kG)) ≤ dimk(J(kH)) + |G| − |H|.

Proposition 1.5.6. [Kar87, Proposition III.1.8] Let N ¢ G with |G : N| = n and p = char(k). Then 30 CHAPTER 1. MODULAR REPRESENTATION THEORY

1. J(kG)n ⊆ kG · J(kN) ⊆ J(kG)

2. If p - n then J(kG) = kG · J(kN) = J(kN)kG and in particular dimk(J(kG)) =

n · dimk(J(kN)).

3. If J(kG) = kG · J(kN) then p - n.

For p-groups, the following upper and lower bounds on the Loewy length of the the group algebra are known:

Proposition 1.5.7. [Kar87, Proposition IV.1.10] Let G be a p-group of order pa. Then

a(p − 1) + 1 ≤ LL(kG) ≤ pa.

Moreover, LL(kG) = a(p − 1) + 1 if and only if G is elementary abelian, and LL(kG) = pa if and only if G is cyclic [Kar87, Theorem IV.3.2] . For a concise characterisation of those p-groups with LL(Z(kG)) ≤ 7, see [Kar87, Section IV.4].

Theorem 1.5.8. [Kar87, Corollary III.2.4] For any groups G1,...,Gs,

s X LL(k(G1 × ... × Gs)) = LL(kGi) − (s − 1). i=1

Example 1.5.9. Suppose G = hgi is cyclic of order pn. Then by Proposition 1.4.15,

i kG is indecomposable and J(kG) = J(Z(kG)) = spank{g − 1} (see Proposition 1.5.10). Since gi − 1 = (g − 1)(1 + g + ... + gi−1), J(kG) = (g − 1)kG and therefore LL(kG) equals the nilpotency of (g − 1) which is pn. s P ni If G = Cpn1 × Cpn2 × ... × Cpns then LL(kG) = p − (s − 1). i=1

1.5.2 Jacobson radical of the centre of the group algebra

P P The map ν : kG → k, g∈G αgg 7→ g∈G αg is a homomorphism called the augmentation map of kG. P P The augmentation ideal I of kG is the set { g∈G αgg | ν( g∈G αgg) = 0}. This is a two-sided ideal. I is nilpotent if and only if G is a p-group; in this case I = J(kG). 1.5. JACOBSON RADICAL AND LOEWY LENGTH 31

Proposition 1.5.10. [Kar87, Proposition III.1.13] Let G be a group such that kG is indecomposable. If C (x1),..., C(xr) are all conjugacy classes distinct from 1G then the elements

Cb(xi) − |C (xi)| · 1G 1 ≤ i ≤ r constitute a k-basis for J(Z(kG)).

Proof. Let π : Z(kG) → k be the restriction of ν to Z(kG). Then π is a homomorphism and hence J(Z(kG)) = ker(π). The elements Cb(xi) − |C (xi)| · 1G are linearly independent and ν Cb(xi) − |C (xi)| · 1G = 0. Hence their linear span has dimension r and is contained in J(Z(kG)). By Theorem 1.5.4, J(Z(kG)) has dimension r, and so the result follows.

Example 1.5.11. Let G = S3 and p = 3. Since G is solvable and Op0 (G) = 1, Theorem 1.4.15 implies that the group algebra kG is indecomposable. The conjugacy classes are

C (1G), C ((12)), C ((123)) with

|C (1G)| = 1, |C ((12))| = 3, |C ((123))| = 2.

Hence by Proposition 1.5.10, {Cb((12)), Cb((123))−2} constitutes a k-basis for J(Z(kG)). Furthermore,

Cb((12))2 ≡ Cb((12)) · (Cb((123)) − 2) ≡ (Cb((123)) − 2)2 ≡ 0 mod J(O)G; hence LL(Z(kG)) = 2.

The following description of J(Z(kG)) is due to Iizuka-Watanabe and K¨ulshammer.

Theorem 1.5.12. [K¨ul91,Equation 59] For every ﬁnite group G,

ˆ expp(G) J(Z(kG)) = {z ∈ Z(kG) | zGp = 0} = {z ∈ Z(kG) | z = 0};

here Gp denotes the set of all elements in G whose order is a power of p and expp(G) denotes the p-part of the exponent of G.

Theorem 1.5.13. [Isa94, Theorem 15.38] If C (g) ∩ CG(Op(G)) = ∅ then Cb(g) ∈ J(Z(FG)). 32 CHAPTER 1. MODULAR REPRESENTATION THEORY

A lot of research has been carried out in trying to give a lower bound on the Loewy length of a block, often in terms of the defect of the block, see for example [Kos14] or [KKS14]. We state two results by K¨ulshammer.

Proposition 1.5.14. [K¨ul82,Korollar K] Let B be a block of kG with defect group D

n and n ∈ N0 with q := p < exp(D); let K(B) be the commutator subspace of B. Then there exists an element j ∈ J(B) with jq ∈/ K(B); in particular (J(B))q 6= 0.

Proposition 1.5.15. [K¨ul82,Korollar L] Let B be a block of kG with defect group D

n q and n ∈ N0 with q := p ≥ exp(D). Then z = 0 for every element z ∈ J(Z(B)).

Passman proved the following upper bound for the Loewy length of the centre of a group algebra.

Theorem 1.5.16. [Pas80] Let char(k) = p > 0 and let |G| = pab with p - b. Then a+1 (J(Z(kG)))(p −1)/(p−1) = 0.

Often, the group algebra is decomposable and hence one is interested in the structure of the Jacobson radical of the centre of a block.

Proposition 1.5.17. [Nav98, Corollary 3.12] If B is a block of G with corresponding block idempotent eB, then

Z(B) = Z(kG) ∩ B = Z(kG)eB is a local algebra with

J(Z(B)) = J(Z(kG)) ∩ B = J(Z(kG))eB.

Remark 1.5.18. By Proposition 1.5.10, the set B := {Cb(xi) − |C (xi)| · 1G | xi ∈ P} forms a basis for J(Z(kG)) if kG is indecomposable. Now consider the principal block

B0 = kGe0 of G. Then

n o D := Cb(xi) − |C (xi)| · 1G e0 | xi ∈ P

is a spanning set for J(Z(kG))e0 = J(Z(B0)).

The following result plays an important role in determining the Loewy length of a block. 1.5. JACOBSON RADICAL AND LOEWY LENGTH 33

Proposition 1.5.19. [KM01, Lemma 4.1(i)]) Suppose that B covers a block b of a normal subgroup H of G. Then LL(b) ≤ LL(B). If p does not divide |G : H| then LL(B) = LL(b).

The result in this proposition does not hold when considering the centre of a block; indeed Loewy lengths of centres are “badly behaved”. This is demonstrated in the following examples (the individual Loewy lengths are discussed in more detail in Chapter 6).

∼ Example 1.5.20. Let N = (C3 × C3) o C8 and G = N o C2 = (C3 × C3) o QD16; ﬁx p = 3. Then 3 does not divide |G : N| and both kN and kG are indecomposable. The Loewy lengths of the centres are

LL(Z(kN)) = 2 < 3 = LL(Z(kG)).

Example 1.5.21. Let N = (C3 × C3) o C4 and G = (C3 × C3) o C8; ﬁx p = 3. Then 3 does not divide |G : N| and both kN and kG are indecomposable. The Loewy lengths of the centres are

LL(Z(kN)) = 3 > 2 = LL(Z(kG)).

Example 1.5.22. Let N = (C3 × C3) o C2 and G = (C3 × C3) o C4; ﬁx p = 3. Then 3 does not divide |G : N| and both kN and kG are indecomposable. The Loewy lengths of the centres are

LL(Z(kN)) = 3 = LL(Z(kG)).

Example 1.5.23. Let G = NMcl(P ), P be a Sylow 5-subgroup of McL and p = 5. Then 5 does not divide |G : P | = 23 · 3, but LL(Z(kP )) > LL(Z(kG)) (note that both group algebras are indecomposable). Therefore the two Loewy lengths are diﬀerent; in fact, the Loewy length of the centre of the block corresponding to the subgroup P is larger than the one corresponding to the group. This is due to the fact that the

Sylow 5-subgroup P has four non-trivial elements in its centre i.e. dimk(Z(P )) = 5. Hence instead of taking various powers of a class sum, we are multiplying an element with itself, meaning that it can’t possibly vanish early on. 34 CHAPTER 1. MODULAR REPRESENTATION THEORY

1.5.3 Tensor Products

We use this subsection to state some results relating to the centre and Jacobson radical of the tensor product of two algebras. These will be important in Section 3.5.

Lemma 1.5.24. Let k be a ﬁeld and A,B ﬁnite dimensional k-algebras. Then

Z(A ⊗k B) = Z(A) ⊗k Z(B).

P Proof. “ ⊇ ” Let x ∈ Z(A), y ∈ Z(B), and t ∈ A ⊗ B. Then t = ai ⊗ bi where ai ∈ A, bi ∈ B.

P t(x ⊗ y) = ( ai ⊗ bi)(x ⊗ y) P = ((ai ⊗ bi)(x ⊗ y)) P P = aix ⊗ biy = xai ⊗ ybi = (x ⊗ y)t

Thus x ⊗ y ∈ Z(A ⊗ B), and all generators of Z(A) ⊗ Z(B) are in Z(A ⊗ B). P “ ⊆ ” We ﬁrst show Z(A ⊗ B) ⊆ Z(A) ⊗ B. Let t ∈ Z(A ⊗ B); then t = ai ⊗ bi. For all a ∈ A, t(a ⊗ 1) = (a ⊗ 1)t, hence

X X (a ⊗ 1)t = aai ⊗ bi = aia ⊗ bi = t(a ⊗ b),

and therefore aai = aia. This shows that ai ∈ Z(A), and so t ∈ Z(A)⊗B. Equivalently, t ∈ A ⊗ Z(B). Finally Z(A ⊗ B) ⊆ (Z(A) ⊗ B) ∩ (A ⊗ Z(B)) = Z(A) ⊗ Z(B).

Proposition 1.5.25. [BZ15, Lemma 5] Let k be an algebraically closed ﬁeld and let A, B be ﬁnite dimensional k-algebras. Then

J(A ⊗k B) = J(A) ⊗k B + A ⊗k J(B).

Proposition 1.5.26. [BZ15, Lemma 9] Let k be an algebraically closed ﬁeld and let m, n be positive integers. Let A, B be ﬁnite dimensional commutative k-algebras. If J n+1(A) = 0 6= J n(A) and J m+1(B) = 0 6= J m(B) (i.e LL(A) = n + 1, LL(B) = m + 1), then

n+m+1 n+m n m J (A ⊗k B) = 0 6= J (A ⊗k B) = J (A) ⊗k J (B). 1.6. THE REYNOLDS IDEAL 35 1.6 The Reynolds ideal

Recall that an element g ∈ G is called a p-element (p0-element) if its order is a power of p (prime to p). Any element g ∈ G can be expressed uniquely as g = gpgp0 = gp0 gp 0 where gp is a p-element and gp0 is a p - element. Therefore for g ∈ G the p-section of g in G is the set of elements h ∈ G such that hp is conjugate to gp. Similarly one can deﬁne the p0-section of g in G. Thus G splits into disjoint p-sections (p0-sections) and each p-section (p0-section) of G splits into disjoint conjugacy classes of G. Moreover, every p0-section of G contains a unique p0-conjugacy class; in particular the number of p0-sections of G is the same as the number of p0-conjugacy classes of G. We denote the

0 set of p -sections of G by Secp0 (G).

Deﬁnition 1.6.1. Let Zp0 (kG) denote the k-subspace of Z(kG) spanned by all the p0-section sums in kG. Reynolds was the ﬁrst to observe that this is an ideal of Z(kG) [Rey72, Theorem 1], hence it is now referred to as the Reynolds ideal in the literature.

ˆ If G is a p-group, Zp0 (kG) is the one dimensional space containing G, and if G is a 0 p -group, Zp0 (kG) = Z(kG). In the case that Op(G) 6= 1 all products in Zp0 (kG) are zero [Rey72, Theorem 4].

2 P Remark 1.6.2. With I1(kG) as in Deﬁnition 1.4.3, I1(kG) = B Z(B) where B ranges over the set of blocks of defect zero in kG [K¨ul91,(82)]. In particular, if kG is

2 indecomposable, then I1(kG) = 0.

We now show that I1(kG) is contained in Soc(kG) = {x ∈ kG | xJ(kG) = 0kG}.

Lemma 1.6.3. The space I1(kG) is an ideal in Z(kG) contained in Soc(kG).

ˆ Proof. By [K¨ul91,Equation (39)], Soc(kG) ∩ Z(kG) = spank{S | S ∈ Secp0 (G)} =: Reynolds ideal (see Deﬁnition 1.6.1). Any conjugacy class of p-defect zero in G is a p0-section of G, therefore we have

I1(kG) ⊆ Zp0 (kG) = Soc(kG) ∩ Z(kG) ⊆ Soc(kG).

The next theorem gives an upper bound on the Loewy length of the centre of a block; unfortunately, in general, the bound is not very tight. 36 CHAPTER 1. MODULAR REPRESENTATION THEORY

Theorem 1.6.4. [Oku81, Theorem] Let e be a block idempotent of kG with corresponding block of defect d. If k is algebraically closed then LL(Z(kGe)) ≤ pd, with equality if and only if kGe is p-nilpotent with a cyclic defect group.

pd−1 Okuyama proves that J(Z(kG)) e ⊆ Zp0 (kG). By the result quoted in the proof of Lemma 1.6.3, Zp0 (kG) ⊆ Soc(kG) and therefore J(Z(kG))Zp0 (kG) = 0. Hence the upper bound on the Loewy length follows.

Theorem 1.6.5. [Oku80, Theorem 2.E] Let B be a p-block of G with corresponding primitive central idempotent e. Then the number of irreducible kG-modules in B, l(B), equals dim((Zp0 (kG))e).

The last fact explains the interest in this speciﬁc subalgebra; l(Bi) is an invariant which plays an important role in many open conjectures. A subspace of Z(kG) closely related to the Reynolds ideal has found attention in the literature in recent years, and is worth mentioning here. Its importance will be discussed further in Remark 2.5.4.

∗ Denote by Zp0 (kG) the k-subspace of Z(kG) spanned by all p-regular conjugacy ∗ ∗ ∗ class sums and set Zp0 (B) = B ∩ Zp0 (kG); in general Zp0 (B) is not multiplicatively closed. Let G be a ﬁnite group, P ∈ Sylp(G), and B a block of G with defect group ∗ ∗ D. Then Zp0 (kG) is a subalgebra if P is abelian [Mey06, Theorem 2] and Zp0 (B) is a subalgebra if D is abelian [FK07, Theorem 2.4] or dihedral [EK08, Theorem 4].

∗ 2 Moreover, in the last case, J(Z20 (B)) = 0 [EK08, Theorem 4]. Furthermore, Meyer ∗ classiﬁes some minimal groups where Zp0 (kG) is not an algebra [Mey06]. ∗ We have dimk(Zp0 (B)) = dimk(Zp0 (B)) = l(B). Chapter 2

Equivalences of blocks

In this chapter, we brieﬂy introduce three levels of equivalence between block algebras which are relevant for the results presented in this thesis: Morita equivalence, derived equivalence and stable equivalence of Morita type. Details are mostly taken from [Bro94], and for further material on these, the reader should refer to for example [Bro94], [KZ98] or [Zim14]. As usual, given any p-block B, k(B) denotes the number of ordinary irreducible characters in B, l(B) denotes the number of modular irreducible characters in B, and d(B) denotes the defect of B. Moreover, the number of ordinary irreducible characters and of modular irreducible characters of G are denoted by k(G) and l(G), respectively. We retain the assumption that (K, O, k) is a p-modular system for G.

2.1 Brauer correspondence

There is a way to connect block theory to local representation theory, which is called the Brauer correspondence. If we ﬁx a p-subgroup D of G then Brauer’s First Main Theorem states:

Theorem 2.1.1 (Brauer’s First Main Theorem). [Nav98, Theorem 4.12] There is a bijection between the set of blocks of G with defect group D and the set of blocks of

NG(D) with defect group D.

The bijection is called the Brauer correspondence and given a block B of G with defect group D, the corresponding block b of NG(D) is called the Brauer correspondent. In

37 38 CHAPTER 2. EQUIVALENCES OF BLOCKS this case one usually writes bG = B. Unless the defect group is a normal subgroup of

G, its normalizer NG(D) is a proper subgroup of G, and so the Brauer correspondent is a block of a strictly smaller group.

In this thesis, we restrict to the principal block of G. In this case, the defect groups are the Sylow p-subgroups of G, and the Brauer correspondent is the principal block of NG(P ) where P is a Sylow p-subgroup. This is a consequence of Brauer’s Third Main Theorem [Nav98, Theorem 6.7].

2.2 Morita equivalence

From now on, we denote by G and H two ﬁnite groups, bye ˆ and fˆ two primitive central idempotents of OG and OH respectively. We set A := OGeˆ and B := OHfˆ. The two blocks A and B are Morita equivalent if the categories of left A-modules, mod(A), and left B-modules, mod(B), are equivalent.

Proposition 2.2.1. [Bro94, Proposition 3.3] Suppose that A and B are Morita equivalent. Then k(A) = k(B), l(A) = l(B) and d(A) = d(B).

View the algebra A as an (A, A)-bimodule. The ring EndA(A)A of its endomorphisms is the centre Z(A) of A. The projective centre Zpr(A) is deﬁned to be the ideal of

Z(A) consisting of those homomorphism from AAA to itself which factor through a projective (A, A)-bimodule. The stable centre is deﬁned as Zst(A) := Z(A)/Zpr(A). Note that by [Bro94, Proposition 2.1], which provides an equivalent formulation

pr for the projective centre, it follows that for A = kG, we have Z (kG) = I1(kG) with

I1(kG) as deﬁned in Deﬁnition 1.4.3.

Proposition 2.2.2. [Bro94, Corollary 3.5] A Morita equivalence between A and B induces an algebra isomorphism between Z(A) and Z(B) which restricts to an isomorphism between Zpr(A) and Zpr(B).

2.3 Derived equivalence

Given two blocks A := OGeˆ and B := OHfˆ we say that A is derived equivalent to B if there exists an equivalence of triangulated categories between the bounded derived 2.3. DERIVED EQUIVALENCE 39 categories Db(mod(A)) and Db(mod(B)). Since the derived category of a block is deﬁned in terms of the module category of that block, then two blocks being Morita equivalent implies they are derived equivalent.

2.3.1 Brou´e’sabelian defect group conjecture

Let A and B be as before. To any generalized character µ of G × H, written µ ∈

ZIrr(K(G × H)), one can associate a linear map Iµ : ZIrr(KH) → ZIrr(KG) by deﬁning for χ ∈ Irr(KH) and g ∈ G,

1 X I (χ)(g) = µ(g, h−1)χ(h). µ |H| h∈H

Deﬁnition 2.3.1. [Bro90] A generalized character µ of G×H is perfect if the following two conditions are satisﬁed: (B1) If µ(g, h) 6= 0 then either g and h are both p-regular, or both are p-singular;

(B2) for all g ∈ G and h ∈ H, |CG(g)|p and |CH (h)|p both divide µ(g, h).

If furthermore the map Iµ deﬁned by µ induces a bijective isometry between ZIrr(KB) and ZIrr(KA), i.e. a bijection which preserves technical conditions upon p- singular elements as deﬁned in [Bro90], then Iµ is said to be a perfect isometry between B and A, and B and A are said to be perfectly isometric.

Brou´e’sPerfect Isometry Conjecture: If B ∈ Bl(G) is a block of G with abelian defect group D and b ∈ Bl(NG(D)) is its Brauer correspondent then there is a perfect isometry between B and b.

Since a derived equivalence between A and B induces a perfect isometry between the two blocks [Bro90, Theorem 3.1], the existence of a perfect isometry in this case is deduced from a stronger statement.

Brou´e’sAbelian Defect Group Conjecture [Bro90, Question 6.2] If B is a block of G with abelian defect group D and b is the Brauer correspondent of B in NG(D), then the categories of the algebras B and b are derived equivalent.

A great amount of representation theory of finite groups has been devoted to proving this conjecture; most notably it has been verified for blocks with cyclic defect groups 40 CHAPTER 2. EQUIVALENCES OF BLOCKS through work of Linckelmann [Lin91], Rickard [Ric89] and Rouquier [Rou98], and for blocks with defect group C2 × C2. The Perfect Isometry Conjecture is of course easier to check for a particular group, and has been verified in far more cases.

The following theorem by Brou´estates that if a perfect isometry exists, it has important consequences; in particular it tells us that the centres are isomorphic.

Theorem 2.3.2. [Bro90, Theorem 1.5][Bro94, Theorem 4.11] Suppose µ induces a perfect isometry between A and B. Then Iµ induces an algebra isomorphism from Z(A) to Z(B) which restricts to an isomorphism between Zpr(A) and Zpr(B). Furthermore, k(A) = k(B), l(A) = l(B) and d(A) = d(B).

Note that a derived equivalence over O gives one over k (the converse is not true in general). Since we are interested in showing that no derived equivalence exists, it is therefore enough to show that the centres are not isomorphic over k.

2.4 Stable equivalence of Morita type

A weaker notion of equivalence is given by a stable equivalence. The stable module category is closely related to the derived category: it is a natural quotient in the sense of triangulated categories. The category has the same objects as mod(A) but the morphisms are equivalence classes: f ∼ g ⇔ f − g factors through a projective module. Hence an equivalence of derived categories induces an equivalence of stable module categories. As we will see later, blocks with trivial intersection defect groups give rise to a particular type of stable equivalence. It is in essence a Morita equivalence up to a projective part; hence the following deﬁnition is made.

Deﬁnition 2.4.1. We say that A := OGeˆ and B := OHfˆ are stably equivalent of Morita type if there exist two bimodules AMB and BNA satisfying the following properties:

1. all of the one-sided modules AM, MB, BN and NA are projective, and

∼ 2. there is an (A, A)-bimodule isomorphism AM ⊗B NA = A ⊕ U for a projective ∼ (A, A)-bimodule U and there is a (B,B)-bimodule isomorphism BN ⊗A MB = 2.5. PROPERTIES OF BLOCKS WITH TI DEFECT GROUPS 41

B ⊕ V for a projective (B,B)-bimodule V .

We remark that M and N induce a Morita equivalence precisely if U and V are zero.

As observed by Rickard, for self-injective algebras, the two notions of derived equivalence and stable equivalence of Morita type are closely related to each other.

Theorem 2.4.2. [Ric91, Corollary 5.5] For self-injective algebras, each derived equivalence induces a stable equivalence of Morita type.

A stable equivalence of Morita type between A and B preserves the numerical invariants as following: k(A) − l(A) = k(B) − l(B) and d(A) = d(B) [Bro94, Proposition 5.3].

Proposition 2.4.3. [Bro94, Proposition 5.4] A stable equivalence of Morita type between A and B induces an algebra isomorphism between Zst(A) and Zst(B).

We are interested in the problem of whether or not this isomorphism can be lifted to an isomorphism between Z(A) and Z(B).

2.5 Properties of blocks with TI defect groups

Recall from Deﬁnition 1.4.9 that a block B is said to have trivial intersection (TI)

g defect groups D if D ∩ D = {1} for all g ∈ G \ NG(D).

Remark 2.5.1. Suppose OGeˆ is a block of G with trivial intersection defect groups D ˆ ˆ and ONf is its Brauer correspondent in ON where N := NG(D). The (ONf, OGeˆ)- bimodule fˆOGeˆ and the (OGe,ˆ ONfˆ)-bimodulee ˆOGfˆ induce a stable equivalence of Morita type between OGeˆ and ONfˆ [KZ98, Section 11.2].

Blocks with TI defect groups have interesting properties and it is worth noting that a number of the open conjectures in modular representation theory have been veriﬁed in this case.

Theorem 2.5.2. [AE05] Suppose B is a block with TI defect groups. Then Dade’s projective conjecture, Robinson’s conjecture, Alperin’s weight conjecture, Alperin-McKay’s conjecture, Isaacs-Navarro’s conjecture and Puig’s nilpotent block conjecture all hold for B. 42 CHAPTER 2. EQUIVALENCES OF BLOCKS

Theorem 2.5.3. [BM90, Theorem 9.2] Let G be a ﬁnite group group with TI Sylow p-subgroup P . Let B be a p-block with defect group P and Brauer correspondent b in

N = NG(P ). Then

• k(B) = k(b);

• k0(B) = k0(b);

• l(B) = l(b);

• k(G) = k(N) + z(G),

where k0(B) denotes the number of ordinary irreducible characters of height zero in B and z(G) denotes the number of p-blocks of G with defect zero.

Blocks with trivial intersection defect groups D are stably equivalent of Morita type to their Brauer correspondent in NG(D), however derived equivalences are not known except in some simple cases. Even more, when Broue’s conjecture does not apply, there are examples when it is known that the principal blocks are not derived equivalent. The ﬁrst example of stably equivalent blocks that are not derived equivalent was given

2 by the principal blocks of the Suzuki group B2(8) and of the normaliser of a Sylow 2-subgroup over a ﬁeld k of characteristic 2. The Sylow 2-subgroups form a TI set, however since the centres of the principal blocks are not isomorphic over O, the two blocks are not derived equivalent [Cli00] (This example is discussed in detail in Section 4.2).

∗ Remark 2.5.4. For blocks with abelian defect groups, the subspace Zp0 (B) introduced in Section 1.6 is invariant under perfect isometries and hence under derived equivalences [FK07]. Unfortunately, as we have seen in Section 1.6, this subspace is, in general, not multiplicatively closed; nevertheless it may still help in investigating derived equivalences. For blocks with trivial intersection defect groups, by Theorem 2.5.3, we know that l(B) = l(b); therefore the dimension of this subspace is equal for both blocks. Hence a ﬁrst step in establishing an isomorphism between the centre of B and the centre of the Brauer correspondent b might be to determine whether such an isomorphism would

∗ ∗ send Zp0 (B) to Zp0 (b). 2.5. NO DERIVED EQUIVALENCE 43

We should also point out that in their two papers K¨ulshammer et.al. ([EK08], [FK07]) also state that, under certain conditions on the group and on the block,

∗ ∗ 2 Zp0 (B) is a subalgebra and (JZp0 (B)) = 0, which might provide a ﬁrst step towards calculating the Loewy length of the centre of the block. However, this is not the approach taken for the results presented in this thesis; rather we consider the centre of a block directly.

2.6 On using the centre to show no derived equivalence exists

We concentrate on the principal blocks, they have the Sylow p-subgroups as their defect groups. Let G be a ﬁnite group and N = NG(P ) where P ∈ Sylp(G); moreover, let B0 and b0 denote the principal blocks of kG and kN respectively. Suppose that the Sylow p-subgroups are trivial intersection. Then by Remark 2.5.1, the two blocks B0 and b0 st ∼ st are stably equivalent of Morita type. Hence by Proposition 2.4.3, Z (B0) = Z (b0). We are interested to ﬁnd out if this isomorphism lifts to an isomorphism between the centres Z(B0) and Z(b0). We now outline our approach. Since Z(B) is a local algebra by Proposition 1.5.17, dim(J(Z(B))) =dim(Z(B))−1.

For B = B0 and b0, use Remark 1.5.18 to ﬁnd a spanning set for J(Z(B)) and determine the Loewy length of Z(B), i.e. the least positive integer n such that J n(Z(B)) = 0. ∼ If LL(Z(B0)) 6= LL(Z(b0)), then Z(B0) 6= Z(b0). 2 2 If LL(Z(B0)) = LL(Z(b0)), consider dim(J (Z(B0))) and dim(J (Z(b0))) instead. 2 2 ∼ If dim(J (Z(B0))) 6= dim(J (Z(b0))), then Z(B0) 6= Z(b0).

By Theorem 2.3.2, there is no perfect isometry between B0 and b0 and therefore there is no derived equivalence between B0(kG) and b0(kN) over k, and hence over O. In the case that the Sylow p-subgroups are abelian, Brou´e’sabelian defect group conjecture predicts a derived equivalence between the two blocks, and hence an isomorphism between the two corresponding centres. Therefore we concentrate on groups with non-abelian, trivial intersection Sylow p-subgroups. Chapter 3

Blocks with trivial intersection defect groups

Throughout this chapter G will denote a finite group and (K, O, k) is a p-modular system (see Definition 1.3.2); in particular, k is an algebraically closed field of characteristic p. Furthermore P denotes a Sylow p-subgroup of G and N = NG(P ) its normaliser in G. We shall concentrate on the principal blocks. Recall from Section 1.4 that the defect groups of the principal blocks are the Sylow p-subgroups. From the classification theorem of finite simple groups, and Theorem 24.1 of Gorenstein-Lyons [GL83, p.307], the sporadic simple groups with non-cyclic TI Sylow p-subgroups are:

• M11 with p = 3;

• McL with p = 5;

• J4 with p = 11.

The motivation for the work presented here comes from a result about the Suzuki

2 2m+1 groups. Cliff [Cli00] proved that for G = B2(q), where q = 2 ≥ 8, the centre of the principal 2-block of kG and the centre of the Brauer correspondent in kN are isomorphic over a field of characteristic 2 (this is further discussed in Chapter 4). In particular, Cliff concluded that an isomorphism in characteristic 2 exists from the following two results:

• the centres Z(kGe0) and Z(kNG(P )) have the same dimension over k;

44 3.1. THE MATHIEU GROUP M11 WITH P = 3 45

• the Jacobson radical squared of each centre is equal to zero.

The ﬁrst result follows from the fact that both blocks contain the same number of irreducible characters which we know to be true for blocks with TI defect groups (see Theorem 2.5.3). Therefore the main work goes into proving the second result. For most groups, the Loewy length of the centre of the principal block is greater

2 2 than two. However, when this happens, the dimensions of J (Z(kGe0)) and J (Z(kN)) often turn out to be diﬀerent; hence despite having the same Loewy length, the centres of the blocks are not isomorphic when P is trivial intersection and not abelian. Due to the sizes of the groups considered, a lot of the computations were carried out in the computer algebra system GAP; the code can be found in Appendix A. Furthermore we refer the reader to the Atlas [CCN+85] for information about the structure of NG(P ) for the groups G considered in this chapter. Note that using Re- mark 1.4.10, together with Lemma 1.4.6 and Theorem 1.4.7, the block decompositions of the group algebras can easily be obtained from the character tables presented in [CCN+85]. Finally, the bases and spanning sets for the centre of a block written down in this chapter follow from Proposition 1.5.10 and Remark 1.5.18.

3.1 The Mathieu group M11 with p = 3

Brou´e’sabelian defect group conjecture has been veriﬁed for the principal block of M11 when p = 3 [Oku98]. Hence we already know that an isomorphism exists. However we determine the structure of the centres and hence explicitly construct such an isomorphism. This also demonstrates in detail the methods and notation used throughout this chapter.

Let G = M11 be the Mathieu group of order 7920, and ﬁx p = 3.

3.1.1 Normaliser of Sylow 3-subgroup

∼ Let P be a Sylow 3-subgroup of G. Then N = NG(P ) = (C3 × C3) o SD16 and

|N| = 144. The group algebra kN consists of only one block, kN = b0, of defect 2, and dim(Z(kN)) = 9. All non-trivial conjugacy classes have class size divisible by 3, except the conjugacy class C (3A) which has size |C (3A)| = 8. Hence a basis for J(Z(kN)) is given by BN = 46 CHAPTER 3. BLOCKS WITH TI DEFECT GROUPS

{Cb(2A), Cb(2B), Cb(3A) + 1, Cb(4A), Cb(4B), Cb(6A), Cb(8A), Cb(8B)} and BN ∪ {1N } is a k-basis of Z(kN).

The character table of NG(P ) is given by:

|CNG(P )(g)| 144 16 12 18 8 4 6 8 8 |C (g)| 1 9 12 8 18 36 24 18 18 |g| 1A 2A 2B 3A 4A 4B 6A 8A 8B

χ1 1 1 1 1 1 −1 1 −1 −1

χ2 1 1 −1 1 1 1 −1 −1 −1

χ3 1 1 −1 1 1 −1 −1 1 1

χ4 1 1 1 1 1 1 1 1 1

χ5 2 −2 0 2 0 0 0 2i −2i

χ6 2 −2 0 2 0 0 0 −2i 2i

χ7 2 2 0 2 −2 0 0 0 0

χ8 8 0 −2 −1 0 0 1 0 0

χ9 8 0 2 −1 0 0 −1 0 0

We use Burnside’s formula, Formula 1.2.2, to calculate the structure constants in the multiplication of two class sums. For example (Cb(2A))2 = 9 · Cb(1A) + 9 · Cb(3A) ≡ 0 mod J(O)N. We ﬁnd that most pairs of conjugacy class sums do in fact multiply to zero

0 0 in kN. For all b, b ∈ BN \{Cb(2B), Cb(6A)} or b ∈ BN \{Cb(2B), Cb(6A)}, b ∈ {Cb(2B), Cb(6A)} we have b · b0 = 0. The following non-zero multiplications occur

(Cb(2B))2 = (Cb(6A))2 = Cb(2A) + Cb(4A), Cb(2B)Cb(6A) = 2 · (Cb(2B))2 = 2 · Cb(2A) + 2 · Cb(4A).

2 (2) Hence a basis for J (Z(kN)) is given by BN = {Cb(2A) + Cb(4A)}. (2) Note that the basis element given for BN does not involve the class sum Cb(2B) or Cb(6A). Therefore Cb(2A) + Cb(4A) · a = 0 for any a ∈ J(Z(kN)) and we can conclude that J 3(Z(kN)) = 0.

Robinson’s formula

There is a more direct approach to prove that the cube of the Jacobson radical must be zero, in other words LL(Z(kN)) = 3. We do not explicitly need to calculate all 3.1. THE MATHIEU GROUP M11 WITH P = 3 47 structure constants in detail if we are just interested in understanding the coeﬃcients in k.

Firstly note that none of the entries in the character table of NG(P ) is divisible by 3, including the character degrees χi(1N ).

Suppose Cb(x), Cb(y), Cb(z) ∈ BN \{1 + Cb(3A)}. Then from the centralizer sizes,

|CN (g)|3 = 3 (if g ∈ C (2B) or C (6A)), and |CN (g)|3 = 1 (if g ∈ C (2A), C (4A), C (4B), C (8A) or C (8B)). Moreover |N|2 = 34 · 24. Hence by Formula 1.2.4, ! X 34 · 24 X u Cb(x)Cb(y)Cb(z) = × Cb(w) 3a · b χ ( ) w i i N where a ∈ {1, 2, 3} and gcd(b, 3) =gcd(u, 3) =gcd(χi(1N ), 3) = 1. Hence if Cb(x), Cb(y),

Cb(z) ∈ BN \{1 + Cb(3A)}, then Cb(x)Cb(y)Cb(z) ≡ 0 mod J(O)N.

Remark. This outline gives a short explanation why we expect the cube of the Ja- cobson radical of the centre of the block to be zero. While this is very useful, we see in the next section that if we want to construct an explicit isomorphism, we need to know how elements in the basis of J(Z(kN)) multiply; in particular we need to calculate J 2(Z(kN)). However note that the idea of the argument described here will form a vital part when considering the Ree groups in Chapter 4.

3.1.2 The group M11

We can observe very similar behaviour in the multiplication of basis elements of the corresponding Jacobson radical for G = M11. The group algebra kG decomposes as one block, B0, of defect 2 and one block, B1, of defect 0; hence dim(Z(kGe0)) = 9. + The character table for M11 can be found in the Atlas [CCN 85], and we follow the notation there.

Let χ9 denote the character of degree 45 belonging to the block B1; lete ˆχ9 denote the corresponding primitive, central block idempotent. By Equation 1.3,

45 eˆχ = 45 − 3Cb(2A) + Cb(4A) − Cb(8A) − Cb(8B) + Cb(11A) + Cb(11B) ∈ Z(KG) 9 7920

Since e0 = 1 − eχ9 , the corresponding principal block idempotent in Z(kG) is given by

e0 = 1 + Cb(4) + 2 · Cb(8A) + 2 · Cb(8B) + Cb(11A) + Cb(11B). 48 CHAPTER 3. BLOCKS WITH TI DEFECT GROUPS

The block idempotents are orthogonal, i.e. e0eχ9 = 0, hence 45 − 3Cb(2A) + Cb(4A) − Cb(8A) − Cb(8B) + Cb(11A) + Cb(11B) e0 = 0 ∈ Z(kGe0).

We can now ﬁx one conjugacy class sum and rearrange:

Cb(8A)e0 = (45 − 3 · Cb(2A) + Cb(4A) − Cb(8B) + Cb(11A) + Cb(11B))e0 (3.1.1)

Let BG = {Cb(2A)e0, (Cb(3A)+1)e0, Cb(4A)e0, Cb(5A)e0, Cb(6A)e0, Cb(8B)e0, Cb(11A)e0,

Cb(11B)e0}. Then BG is a k-basis for J(Z(kGe0)) and BG ∪ {e0} is a k-basis of

Z(kGe0). 0 Most pairs of conjugacy class sums multiply to zero: for all b, b ∈ BG \{Cb(2A)e0, 0 0 Cb(6A)e0} or b ∈ BG \{Cb(2A)e0, Cb(6A)e0}, b ∈ {Cb(2A)e0, Cb(6A)e0} we have b·b = 0, and the following non-zero multiplications occur

2 2 (Cb(2A)e0) = (Cb(6A)e0) = Cb(4A)e0 + 2 · Cb(5A)e0 2 Cb(2A)e0Cb(6A)e0 = 2 · (Cb(2B)e0) = 2 · Cb(4A)e0 + Cb(5A)e0.

Note that Cb(4A)e0 + 2 · Cb(5A)e0 = 2 · Cb(4A) + 2 · Cb(5A) + 2 · Cb(8A) + 2 · Cb(8B) + 2 (2) Cb(11A)+Cb(11B) 6= 0. Hence a basis for J (Z(kGe0)) is given by BG = {Cb(4A)e0+2· (2) Cb(5A)e0}. As the basis element given for BG does not involve the class sums Cb(2A) or Cb(6A), it follows that Cb(4A)e0 + 2 · Cb(5A)e0 · a = 0 where a ∈ J(Z(kGe0)). 3 Therefore J (Z(kGe0)) = 0.

Combining both groups

It remains to write down an explicit isomorphism between the two centres Z(B0) and Z(kN). Note that we have to be careful in matching up the basis elements, but since we know exactly how the elements in both bases multiply together, it is still straight forward. The map f : Z(kN) → Z(kGe0) given by f(1N ) = 1Ge0 and      Cb(2A)   Cb(8B)e0           Cb(2B)   Cb(2A)e0           1 + Cb(3A)   (1 + Cb(3A))e0           Cb(4A)   Cb(4A)e0  f : −→  Cb(4B)   Cb(5A)e0           Cb(6A)   Cb(6A)e0           Cb(8A)   Cb(11A)e0           Cb(8B)   Cb(11B)e0  3.2. THE MCLAUGHLIN GROUP MCL, AND AUT(MCL), WITH P = 5 49 is an isomorphism between the centre of the group algebra kN and the centre of the ∼ principal block of M11, i.e. Z(kN) =f Z(kGe0).

3.2 The McLaughlin group McL, and Aut(McL), with p = 5

We next take a look at the McLaughlin group, a sporadic simple group of order 27 · 36 · 53 · 7 · 11 and ﬁx the prime p = 5.

3.2.1 Normaliser of Sylow 5-subgroup

Let P be a Sylow 5-subgroup of McL; note that P is not abelian. The normaliser ∼ N = NG(P ) = (((C5 o C5) o C5) o C3) o C8 has size 3000 and splits into 19 conjugacy classes.

The group algebra kN is the principal block, kN = b0, of defect 3, and dim(Z(kN)) = 19. The character table can easily be obtained from the GAP character table library using the command gap> CharacterTable(”McLN5”); All non-trivial conjugacy classes have class size divisible by 5 except C (5A) which has class size |C (5A)| = 4; hence a basis for J(Z(kN)) is given by

BN = {Cb(x) | x ∈ P, x 6= 1N , x 6∈ C (5A)} ∪ {Cb(5A) + 1}.

0 There exist basis elements b, b ∈ BN such that

b · b0 6= 0; in particular the following distinct non-zero multiplications occur

Cb(3A) · Cb(3A) = Cb(3A) + Cb(15A) + Cb(15B); Cb(5B) · Cb(10A) = Cb(2A) + Cb(10A); Cb(2A) · Cb(3A) = Cb(6A) + Cb(30A) + Cb(30B).

Any other pair of conjugacy class sums in BN either multiplies to zero in kN or is a non-zero multiple of the three given multiplications. Hence J 2(Z(kN)) has dimension 3 and a basis given by

(2) BN = {Cb(3A) + Cb(15A) + Cb(15B), Cb(2A) + Cb(10A), Cb(6A) + Cb(30A) + Cb(30B)}. 50 CHAPTER 3. BLOCKS WITH TI DEFECT GROUPS

Next we need to establish whether J 3(Z(kN)) = 0. Due to the size of the group, we

0 00 0 00 use GAP to explicitly calculate that for all b, b , b ∈ BN we have b · b · b = 0. Hence J 3(Z(kN)) = 0 and so LL(Z(kN)) = 3.

3.2.2 The group McL

In characteristic 5, the group algebra of the McLaughlin group decomposes into 6 blocks: the principal block, B0, of defect 3 and ﬁve blocks of defect zero. By a result of Blau and Michler, stated in Theorem 2.5.3, dim(Z(B0)) = 19.

The ﬁrst thing to calculate is the block idempotent for the principal block, e0. Using Equation 1.3, we calculate

1 eˆ0 = |G| 646049875 − 1388125 · Cb(2A) − 826250 · Cb(3A) − 33125 · Cb(3B) +40375 · Cb(4A) + 68750 · Cb(6A) + 6875 · Cb(6B) − 3750 · Cb(7A) −3750 · Cb(7B) + 9625 · Cb(8A) − 6125 · Cb(9A) − 6125 · Cb(9B) −6250 · Cb(11A) − 6250 · Cb(11B) − 19250 · Cb(12A) + 12750 · Cb(14A) +12750 · Cb(14B)

As a simple check, note that this answer agrees with the result by K¨ulshammer,The- orem 1.3.4, since none of the conjugacy classes appearing in the block idempotent contain elements of order divisible by 5. All non-trivial conjugacy classes of McL have class size divisible by 5, except for C (5A) which has size |C (5A)| = 1197504. Hence consider the set

DG = {Cb(x)e0 | x ∈ P, x 6= 1G, x 6∈ C (5A)} ∪ {(Cb(5A) + 1)e0}.

This set is not linearly independent, and hence not a basis for J(Z(kGe0)). However it is clearly a spanning set, which is enough for our calculations.

0 00 In GAP we can calculate that for all b, b , b ∈ DG we have

b · b0 · b00 = 0.

0 At the same time note that there exist elements b, b ∈ DG such that

b · b0 6= 0.

2 More precisely, we calculate in GAP that dim(J (Z(kGe0))) = 4. The calculations above lead to the following theorem. 3.2. THE MCLAUGHLIN GROUP MCL, AND AUT(MCL), WITH P = 5 51

Theorem 3.2.1. Let G = McL, N = NG(P ) and k an algebraically closed ﬁeld of characteristic p = 5. Then LL(Z(kGe0)) = LL(Z(kN)) = 3. Moreover,

2 2 dim(J (Z(kGe0))) = 4 6= 3 = dim(J (Z(kN))),

∼ and therefore Z(kGe0) 6= Z(kN).

3.2.3 The group Aut(McL)

Let G = Aut(McL) ∼= McL.2. Then |G| = 1796256000 and G has 33 conjugacy classes. The group algebra kG decomposes into 7 blocks: the principal block B0 of defect 3 and 6 blocks of defect zero. As usual, b0 denotes the principal block of kNG(P ). Using the same methods as those for McL, the following results are obtained:

Block B Defect δ(B) k(B) LL(Z(B)) dim(J 2(Z(B)))

B0 3 (C5 × C5) o C5 26 3 5

b0 3 (C5 × C5) o C5 26 3 4

Similarly to the result in the McLaughlin group, we cannot have an isomorphism of the centres.

Theorem 3.2.2. Let G = Aut(McL), N = NG(P ) and k a ﬁeld of characteristic p = 5. Then

2 2 dim(J (Z(kGe0))) = 5 6= 4 = dim(J (Z(kN))), ∼ and therefore Z(kGe0) 6= Z(kN). 52 CHAPTER 3. BLOCKS WITH TI DEFECT GROUPS

3.3 The Janko group J4 with p = 11

In this section we consider the Janko group J4, a sporadic simple group of order 221 · 33 · 5 · 7 · 113 · 23 · 29 · 31 · 37 · 43, and ﬁx the prime p = 11.

3.3.1 Normaliser of Sylow 11-subgroup

Let P be a Sylow 11-subgroup of J4. The normaliser N = NG(P ) has size 319440 and 49 conjugacy classes. The group algebra kN is indecomposable and so there is only one block kN = b0, of defect 3, and dim(Z(kN)) = 49. The character table can be obtained from the GAP character table library [GAP]. All non-trivial conjugacy classes have class size divisible by 11 except C (11A) which has class size |C (11A)| = 10. Hence a basis for J(Z(kN)) is given by

BN = {Cb(x) | x ∈ P, x 6= 1N , x 6∈ C (11A)} ∪ {Cb(11A) + 1}.

0 00 In GAP we can calculate that for all b, b , b ∈ BN we have

b · b0 · b00 = 0.

0 At the same time note that there exist basis elements b, b ∈ BN such that

b · b0 6= 0.

For example take Cb(2A) · Cb(4A) = Cb(4A) + Cb(44A) (there are many others). More precisely, we calculate in GAP that dim(J 2(Z(kN))) = 4.

3.3.2 The group J4

In characteristic 11, the group algebra kJ4 decomposes into 14 blocks: the principal block, B0, of defect 3 and 13 blocks of defect zero. From the character table [CCN+85], or by using a result of Blau and Michler as stated in Theorem 2.5.3, we have dim(Z(B0)) = 49. All non-trivial conjugacy classes have class size divisible by 11 except C (11A) which has class size |C (11A)| = 2716490453483520. Hence a spanning set for J(Z(kGe0)) is given by

DG = {Cb(x)e0 | x ∈ P, x 6= 1G, x 6∈ C (11A} ∪ {(Cb(11A) + 1)e0}. 3.3. THE JANKO GROUP J4 WITH P = 11 53

1 0 00 In GAP we can calculate that for all b, b , b ∈ DG we have

b · b0 · b00 = 0.

0 At the same time note that there exist elements b, b ∈ DG such that

b · b0 6= 0.

2 More precisely, we calculate in GAP that dim(J (Z(kGe0))) = 5. Hence we get the following theorem.

Theorem 3.3.1. Let G = J4, N = NG(P ) and k an algebraically closed ﬁeld of characteristic p = 11. Then LL(Z(kGe0)) = LL(Z(kN)) = 3. Moreover,

2 2 dim(J (Z(kGe0))) = 5 6= 4 = dim(J (Z(kN))),

∼ and therefore Z(kGe0) 6= Z(kN).

1This computation took over two days to ﬁnish. 54 CHAPTER 3. BLOCKS WITH TI DEFECT GROUPS 3.4 The projective special unitary groups

In this section we consider some projective special unitary groups. We will calculate some block theoretic properties which are required for consideration of our question. The focus lies on the following groups: PSU(3, 3), PSU(3, 3).2, PSU(3, 9), PSU(3, 4),

PSU(3, 8), PSU(3, 5), PSU(3, 5).2, PSU(3, 5).3, PSU(3, 5).S3 and PSU(3, 7).

3.4.1 The group PSU(3, 3) with p = 3

We ﬁrst consider the projective special unitary group PSU(3, 3) ∼= SU(3, 3), a simple group of order 6048 and ﬁx the prime p = 3.

Normaliser of Sylow 3-subgroup

∼ Let P be a Sylow 3-subgroup of PSU(3, 3). The normaliser N = NG(P ) = ((C3 ×

C3) o C3) o C8 has size 216 and splits into 13 conjugacy classes.

The group algebra kN is the principal block, kN = b0, of defect 3, and dim(Z(kN)) = 13. The character table can be obtained from [GAP]; we get the following conjugacy classes with corresponding conjugacy class sizes:

C (g) 1A 3A 2A 6A 8A 4A 12A 8B 8C 4B 12B 8D 3B |C (g)| 1 2 9 18 27 9 18 27 27 9 18 27 24

All non-trivial conjugacy classes have class size divisible by 3 except C (3A) which has size |C (3A)| = 2. Hence a basis for J(Z(kN)) is given by

BN = {Cb(x) | x ∈ P, x 6= 1N , x 6∈ C (3A)} ∪ {Cb(3A) + 1}.

0 00 From calculations in GAP it follows that for all b, b , b ∈ BN ,

b · b0 · b00 = 0.

At the same time the following distinct non-zero multiplications occur

Cb(2A) · Cb(4A) = Cb(4B) + Cb(12B) Cb(2A) · Cb(4B) = Cb(4A) + Cb(12A) Cb(4B) · Cb(4B) = Cb(2A) + Cb(6A) 3.4. THE PROJECTIVE SPECIAL UNITARY GROUPS 55

Any other pair of conjugacy class sums either multiplies to zero in kN or is a non-zero multiple of the three given multiplications. Hence J 2(Z(kN)) has dimension 3 and a basis given by

(2) BN = {Cb(4A) + Cb(12A), Cb(4B) + Cb(12B), Cb(2A) + Cb(6A)}.

The group PSU(3, 3)

In characteristic 3, the group algebra kG decomposes into 2 blocks: the principal block of defect 3 and one block of defect zero. By a result of Blau and Michler, stated in Theorem 2.5.3, we have dim(Z(kGe0)) = 13. Furthermore the principal block idempotent is given by

1 eˆ0 = 6048 5319 − 81 · Cb(2A) − 81 · Cb(4A) − 81 · Cb(4B) + 27 · Cb(4C) +27 · Cb(7A) + 27 · Cb(7B) − 27 · Cb(8A) − 27 · Cb(8B)

The group has the following conjugacy classes with corresponding conjugacy class sizes:

C (g) 1A 2A 3A 3B 4A 4B 4C 6A 7A 7B 8A 8B 12A 12B |C (g)| 1 63 56 672 63 63 378 504 864 864 756 756 504 504

All non-trivial conjugacy classes have class size divisible by 3 except C (3A) which has size |C (3A)| = 56. Hence a spanning set for J(Z(kGe0)) is given by

DG = {Cb(x)e0 | x ∈ P, x 6= 1G, x 6∈ C (3A)} ∪ {(Cb(3A) + 1)e0}.

0 00 In GAP we can calculate that for all b, b , b ∈ DG we have

b · b0 · b00 = 0.

Moreover, the following distinct non-zero multiplications occur

(Cb(2A) · Cb(2A))e0 = 2 · Cb(7A) + 2 · Cb(7B) + Cb(8A) + Cb(8B)

(Cb(2A) · Cb(4B))e0 = Cb(4A) + 2 · Cb(7A) + 2 · Cb(7B) + Cb(8B) + Cb(12A)

(Cb(2A) · Cb(3B))e0 = 2 · Cb(2A) + Cb(4C) + 2 · Cb(6A) + Cb(7A) + Cb(7B) +2 · Cb(8A) + 2 · Cb(8B)

(Cb(2A) · Cb(12A))e0 = Cb(4B) + Cb(7A) + Cb(7B) + 2 · Cb(8A) + Cb(12B)

Any other pair either multiplies to zero in kGe0 or is a non-zero multiple of the four 2 given multiplications. Hence J (Z(kGe0)) has dimension 4. 56 CHAPTER 3. BLOCKS WITH TI DEFECT GROUPS

Summary of block information:

Block B Defect δ(B) k(B) LL(Z(B)) dim(J 2(Z(B)))

B0 3 (C3 × C3) o C3 13 3 4

b0 3 (C3 × C3) o C3 13 3 3

3.4.2 The group PSU(3, 3) : 2 with p = 3

The group G ∼= PSU(3, 3) : 2 has order 12096 and 16 conjugacy classes; ﬁx the prime p = 3. Using similar methods to before, the following block information can be obtained:

Block B Defect δ(B) k(B) LL(Z(B)) dim(J 2(Z(B)))

B0 3 (C3 × C3) o C3 14 3 4

b0 3 (C3 × C3) o C3 14 3 3

Theorem 3.4.1. Let G = PSU(3, 3) or PSU(3, 3) : 2, N = NG(P ) where P ∈

Syl3(G) and k an algebraically closed ﬁeld of characteristic p = 3. Then

2 2 dim(J (Z(kGe0))) > dim(J (Z(kN))), and therefore Z(kGe0) Z(kN).

3.4.3 The group PSU(3, 9) with p = 3

Consider the projective special unitary group PSU(3, 9), a group of order 42573600 with 92 conjugacy classes, and ﬁx the prime p = 3. The group has too many conjugacy classes to compute the Loewy lengths, therefore we only calculate dim(J 2(Z(B))). Block information:

Block B Defect δ(B) k(B) dim(J 2(Z(B)))

B0 6 D 91 10

b0 6 D 91 9 ∼ where D = ((C3 × C3 × C3 × C3) o C3) o C3.

3.4.4 The group PSU(3, 4) with p = 2

Consider the projective special unitary group PSU(3, 4), a group of order 62400 with 22 conjugacy classes, and ﬁx the prime p = 2. 3.4. THE PROJECTIVE SPECIAL UNITARY GROUPS 57

Block information: Block B Defect δ(B) k(B) LL(Z(B)) dim(J 2(Z(B)))

B0 6 D 21 3 5

b0 6 D 21 3 4 ∼ where D = (C2 × C2).(C2 × C2 × C2 × C2).

3.4.5 The group PSU(3, 8) with p = 2

Take the projective special unitary group PSU(3, 8), a group of order 5515776 with 28 conjugacy classes, and ﬁx the prime p = 2. Block information: Block B Defect δ(B) k(B) LL(Z(B)) dim(J 2(Z(B)))

B0 9 D 27 3 3

b0 9 D 27 3 2 ∼ where D = (C2 × C2 × C2).(C2 × C2 × C2 × C2 × C2 × C2).

3.4.6 The group PSU(3, 5) with p = 5

Consider the projective special unitary group PSU(3, 5), a group of order 126000 with 14 conjugacy classes, and ﬁx the prime p = 5. Block information: Block B Defect δ(B) k(B) LL(Z(B)) dim(J 2(Z(B)))

B0 3 (C5 × C5) o C5 13 3 2

b0 3 (C5 × C5) o C5 13 3 1

3.4.7 The group PSU(3, 5) : 2 with p = 5

Consider a subgroup of the automorphism group of the projective special unitary group PSU(3, 5). The group G ∼= PSU(3, 5) : 2 has order 252000 and 19 conjugacy classes; ∼ ﬁx the prime p = 5. Note that N = (((C5 × C5) o C5) o C8) o C2 and |NG(P )| = 2000. Block information: Block B Defect δ(B) k(B) LL(Z(B)) dim(J 2(Z(B)))

B0 3 (C5 × C5) o C5 17 3 3

b0 3 (C5 × C5) o C5 17 3 2 58 CHAPTER 3. BLOCKS WITH TI DEFECT GROUPS

3.4.8 The group PSU(3, 5) : 3 with p = 5

Next, we consider another subgroup of the automorphism group of the projective special unitary group PSU(3, 5). The group G ∼= PSU(3, 5) : 3 has order 378000 and ∼ 34 conjugacy classes; ﬁx the prime p = 5. Note that N = (((C5 × C5) o C5) o C8) o C3 and |NG(P )| = 3000. Block information:

Block B Defect δ(B) k(B) LL(Z(B)) dim(J 2(Z(B)))

B0 3 (C5 × C5) o C5 31 3 6

b0 3 (C5 × C5) o C5 31 3 5

3.4.9 The group PSU(3, 5) : S3 with p = 5

In this section we consider the automorphism group of the projective special unitary ∼ group PSU(3, 5). The group G = PSU(3, 5) : S3 has order 756000 and 29 conjugacy ∼ classes; ﬁx the prime p = 5. Note that N = ((((C5 × C5) o C5) o C8) o C3) o C2 and

|NG(P )| = 6000. Block information:

Block B Defect δ(B) k(B) LL(Z(B)) dim(J 2(Z(B)))

B0 3 (C5 × C5) o C5 26 3 5

b0 3 (C5 × C5) o C5 26 3 4

Theorem 3.4.2. Let G = PSU(3, 5), PSU(3, 5) : C2, PSU(3, 5) : C3 or PSU(3, 5) :

S3; let N = NG(P ) where P ∈ Syl5(G) and k is an algebraically closed ﬁeld of characteristic p = 5. Then

2 2 dim(J (Z(kGe0))) > dim(J (Z(kN))),

∼ and so Z(kGe0) 6= Z(kN).

3.4.10 The group PSU(3, 7) with p = 7

Consider the projective special unitary group PSU(3, 7), a group of order 5663616 with 58 conjugacy classes, and ﬁx the prime p = 7. 3.4. THE PROJECTIVE SPECIAL UNITARY GROUPS 59

Block information:

Block B Defect δ(B) k(B) LL(Z(B)) dim(J 2(Z(B)))

B0 3 (C7 × C7) o C7 57 3 8

b0 3 (C7 × C7) o C7 57 3 7

3.4.11 Remarks

The calculations regarding PSU(3, pr) were done independently by Bouc and Zimmer- mann in a recent paper [BZ15]. Motivated by a question of Rickard, which we discuss in Section 3.5, the authors state the same results for the principal p-block of the group PSU(3, pr) and its Brauer correspondent for pr ∈ {3, 4, 5, 7, 8}. We make the observation that the examples give rise to the following conjecture.

r Conjecture 3.4.3. [BZ15, Remark 15] Let G = PSU(3, p ), B0 the principal block of kG, and b0 the Brauer correspondent of B0 in kNG(P ). Then

2 2 dim(J (Z(B0))) = 1 + dim(J (Z(b0))).

The conjecture has the following consequence.

r Conjecture 3.4.4. Let G = PSU(3, p ), B0 the principal block of kG, and b0 the

Brauer correspondent of B0 in kNG(P ). Then

∼ Z(B0) 6= Z(b0).

2 In their paper, Bouc and Zimmermann explicitly calculate dim(J (Z(b0))) and the Lowey length for the normaliser:

Theorem 3.4.5. [BZ15, Theorem 41] Let k be a ﬁeld of characteristic p; let γ = gcd(pr + 1, 3). Then

q2+q 1. The dimension of J(Z(kN)) is equal to γ + γ − 1.

2 pr+1 2. The dimension of J (Z(kN)) is equal to γ − 1.

3. The cube J 3(Z(kN)) of J(Z(kN)) is equal to 0. 60 CHAPTER 3. BLOCKS WITH TI DEFECT GROUPS 3.5 A question of Rickard

In [Ric98], Rickard asks the following question.

Question 3.5.1. Suppose Λ, Λ, Γ, Γ are connected self-injective k-algebras with Λ stably equivalent to Γ and Λ stably equivalent to Γ. Must Λ⊗k Λ and Γ⊗k Γ be stably equivalent? What if the stable equivalences are of Morita type? Or if the algebras are all block algebras?

We will explain how the results given in this chapter provide counterexamples to Rickard’s question. Let k be an algebraically closed ﬁeld of characteristic p. Then the symmetric k-algebras B0 and b0 are stably equivalent of Morita type, but B0 ⊗k p p k[X]/X and b0 ⊗k k[X]/X are not stably equivalent of Morita type. The proof of this result is discussed in detail in [BZ15] and we will summarise the argument here. Firstly, given a ﬁnite dimensional k-algebra A, we need to compute the Loewy

p length of Z(A ⊗k k[X]/X )

Lemma 3.5.2. Let A be a ﬁnite dimensional k-Algebra. If J n(Z(A)) 6= 0 = J n+1(Z(A)) then

n+p−1 p n p−1 p 0 6= J (Z(A ⊗k k[X]/X )) = J (Z(A)) ⊗k X k[X]/X and

n+p p J (Z(A ⊗k k[X]/X )) = 0.

Proof. [BZ15, Lemma 11] Using Lemma 1.5.24 and Proposition 1.5.26 we establish the following equalities.

n+p−1 p n+p−1 p J (Z(A ⊗k k[X]/X )) = J (Z(A) ⊗k Z(k[X]/X )) n+p−1 p = J (Z(A) ⊗k k[X]/X ) n p−1 p = J (Z(A)) ⊗k J (k[X]/X ) n p−1 p = J (Z(A)) ⊗k X k[X]/X 6= 0

By Proposition 1.5.26, and using the fact that LL(Z(k[X]/Xp)) = LL(k[X]/Xp) = p, we have

(n+p−1)+1 p J (Z(A) ⊗k (k[X]/X )) = 0. 3.5. A QUESTION OF RICKARD 61

Theorem 3.5.3. Let k be an algebraically closed ﬁeld of characteristic p > 0 and let A and B be two ﬁnite dimensional, symmetric k-algebras and let n, m ∈ N such that J n(Z(A)) 6= 0 = J n+1(Z(A)) and J m(Z(B)) 6= 0 = J m+1(Z(B)). If n 6= m or if

n m p p n = m and dimk(J (Z(A))) 6= dimk(J (Z(B))) then A⊗k k[X]/X and B⊗k k[X]/X are not stably equivalent of Morita type.

Proof. [BZ15, Corollary 12] If n 6= m then

p p LL(Z(A ⊗k k[X]/X )) = n + p 6= m + p = LL(Z(B ⊗k k[X]/X ))

p ∼ p and so Z(A ⊗k k[X]/X ) 6= Z(B ⊗k k[X]/X ). n m If dimk(J (Z(A))) 6= dimk(J (Z(B))) then

n+p−1 p n p−1 p dim(J (Z(A ⊗k k[X]/X ))) = dim(J (Z(A)) ⊗k X k[X]/X )

n p−1 p n+p−1 p 6= dim(J (Z(B)) ⊗k X k[X]/X ) = dim(J (Z(B ⊗k k[X]/X ))).

p ∼ p Hence the centres are again not isomorphic, Z(A ⊗k k[X]/X ) 6= Z(B ⊗k k[X]/X ).

Let CA denote the Cartan matrix of A and rankp(CA) its rank as a matrix over pr p k. Then since dimk(Z (A)) = rankp(CA) [BZ15, Theorem 3] and CA⊗kk[X]/X = pr p pr p · CA [BZ15, Lemma 7], we have that Z (A ⊗k k[X]/X ) = 0. Similarly, Z (B ⊗k p st pr p k[X]/X ) = 0. By deﬁnition, Z (A) = Z(A)/Z (A); hence Z(A ⊗k k[X]/X ) = st p p st p Z (A ⊗k k[X]/X ) and Z(B ⊗k k[X]/X ) = Z (B ⊗k k[X]/X ). Finally by Theorem 2.4.3, the stable centre is invariant under stable equivalence of Morita type, and so we get the result.

Let G be a group with trivial intersection Sylow p-subgroups; for example take

G = McL and P ∈ Syl5(G). In this case the principal blocks B0(G) and b0(kNG(P )) p ∼ are stably equivalent of Morita type. However by Theorem 3.2.1, Z(B0 ⊗k k[X]/X ) 6= p p p Z(b0⊗kk[X]/X ) and hence B0⊗kk[X]/X and b0⊗kk[X]/X are not stably equivalent of Morita type, giving another negative example to Rickard’s question. 62 CHAPTER 3. BLOCKS WITH TI DEFECT GROUPS 3.6 On the existence of perfect isometries in blocks with TI defect groups

Throughout this section, we continue to concentrate on the principal blocks. Local representation theory studies the connection between kG and kNG(P ); Brou´e’sabelian defect group conjecture (ADGC) predicts the existence of a derived equivalence between the blocks B0(kG) and b0(kNG(P )) if the Sylow p-subgroups of G are abelian. We investigate the question of whether a derived equivalence can exist when the Sylow p-subgroups are non-abelian, trivial intersection (TI) subgroups of G. Let G be a ﬁnite group with trivial intersection Sylow p-subgroups P . Since we are interested in comparing the principal block algebras of G and NG(P ), we may assume that P is not normal in G. Since Op(G) is the intersection of all Sylow p-subgroups of G, we have Op(G) = 1. The following lemma allows us to further assume that

Op0 (G) = 1.

Lemma 3.6.1. Let P ∈ Sylp(G), N = Op0 (G), G = G/N and P = P N/N ∈ Sylp(G). Then N (P )N N (P ) = G . G N

NG(P )N Proof. It is clear that NG(P ) ⊇ N , therefore it remains to prove the opposite inclusion.

By deﬁnition, NG(P ) = {xN ∈ G | xP = P x} = {x ∈ G | xP = P x}/N =: H/N. Then H = {x ∈ G | x−1P Nx = PN} = {x ∈ G | (x−1P x)N = PN} = {x ∈ G | P xN = PN}.

It is enough to show H ⊆ NG(P )N. x x Let x ∈ G such that P N = PN. Then P and P are in Sylp(PN). Hence there x pn n −1 exists an element pn ∈ PN such that P = P = P . It follows that xn ∈ NG(P ) and therefore x ∈ NG(P )N.

Remark 3.6.2. Let G be a ﬁnite group and H a normal p0-subgroup of G. The

Fong-Reynolds reduction ([Fon61], [Rey63]) tells us that B0(G) is Morita equivalent to B0(G/H), denoted B0(G) ∼M B0(G/H). Hence taking H = NG(P ) ∩ Op0 (G) it 3.6. PERFECT ISOMETRIES IN TI BLOCKS 63 follows that

NG(P ) B0(NG(P )) ∼M B0 by F-R reduction NG(P )∩Op0 (G) ∼ NG(P )Op0 (G) = B0 by 2nd Isomorphism Theorem Op0 (G)

= B0(NG(P )) by Lemma 3.6.1, and B0(G) ∼M B0(G/Op0 (G)). Since Morita equivalent blocks have isomorphic centres

(Proposition 2.2.2), Z(B0(G)) is isomorphic to Z(B0(NG(P )) if and only if Z(B0(G)) is isomorphic to Z(B0(NG(P ))).

From now on, it will be assumed that Op(G) = Op0 (G) = 1.

The p-local rank, plr(G), of G is deﬁned to be the length of a longest chain in the set of radical p-chains of G (for details see [Eat01] or [Rob96]); plr(G) = 0 if and only if G has a normal Sylow p-subgroup. Moreover, if plr(G) > 0, then plr(G) = 1 if and only if G/Op(G) has TI Sylow p-subgroups [Rob96, Lemma 7.1]. Hence for G with the assumptions above, plr(G) = 1. In particular, the following lemma can be applied to G.

Lemma 3.6.3. [Eat01, Lemma 2.4] Let G be a ﬁnite group with plr(G) = 1 and

Op(G) = Op0 (G) = 1. Then there is a unique non-trivial minimal normal subgroup S of G. Furthermore, S is non-abelian simple, plr(S) = 1 and G is isomorphic to a subgroup of the automorphism group of S.

Lemma 3.6.4. [Eat01, Lemma 3.2] Let p be a prime and S be as in Lemma 3.6.3. Then (p, S) is one of the following:

2 2m+1 (a) (2, B2(2 )), m ≥ 1;

2 2m+1 2 0 (b) (3, G2(3 )), (3,PSL(3, 4)), (3, G2(3) ), (3,M11), m ≥ 1;

2 5 2 0 (c) (5, B2(2 )), (5, F4(2) ), (5, McL);

(d) (11,J4);

(e) (p, P SL(2, pm)), (p, P SU(3, pm)), m ≥ 1.

The following lemma appears in [Eat01]. However the proof presented there is incomplete as the group PSL(3, 4) was not covered; we include a short proof here for completeness of the statement. 64 CHAPTER 3. BLOCKS WITH TI DEFECT GROUPS

Lemma 3.6.5. Let G and S be as in Lemma 3.6.3. Then gcd(p, [G : S]) = 1 except

2 0 2 5 when (p, S) = (3, G2(3) ) or (5, B2(2 )).

Proof. Suppose (p, S) 6= (3,PSL(3, 4)). Then the result follows by [Eat01, Lemma 3.3]. ∼ Suppose S = PSL(3, 4) and S ¡ G such that [G : S] = 3. Let P ∈ Syl3(G) and + take an element x ∈ P, x 6∈ S. By [CCN 85], |CG(x)| is divisible by 5 or 7. Note that since NG(P ∩S) = NG(P ∩S) ∼= NG(P ∩S)S ≤ G , we must have that NS (P ∩S) NG(P ∩S)∩S S S ∼ [NG(P ∩ S): NS(P ∩ S)] divides 3. Moreover, NS(P ∩ S) = (P ∩ S) o Q8 which 3 3 implies that |NG(P ∩ S)| divides 3 · 2 . Finally since NG(P ) ≤ NG(P ∩ S), then gcd(5, |NG(P )|) = 1 = gcd(7, |NG(P )|). If P is trivial intersection, then by Corollary

4.1.2, CG(x) ≤ NG(P ), leading to a contradiction. Hence if G has trivial intersection Sylow p-subgroups, then G is a p0-extension of S.

Motivated by the results in this chapter, and the results presented in Chapter 4, we make the following conjecture.

Conjecture 3.6.6. Let G be a ﬁnite group and B = B0(G) with TI defect groups D. G Let b ∈ Bl(NG(D)) such that b = B. Then for p = 2, B is derived equivalent to b if and only if D is abelian or generalised quater- nion;

∼ 1+2 1+2 for p > 2, B is derived equivalent to b if and only if D is abelian or D = 3− , 5− .

∼ 1+2 1+2 Since the existence of a derived equivalence for the cases where D = 3− or 5− is still an open question, we state and prove the perfect isometry version of Conjecture 3.6.6 for the principal blocks in characteristic p = 3 and 5 when D is “small”. The main obstacle in proving the conjecture in full generality comes from the projective special unitary groups. A small number of individual cases were considered in Section 3.4, however further research extending the results presented in [BZ15] is required to establish the non-existence of a perfect isometry for PSU(3, pn) for an arbitrary prime p and integer n. In particular, our bound on the sizes of the defect groups considered arises from this constraint. In addition, the question of what happens in the automorphism groups of the Suzuki and Ree groups is still open. This is also the 3.6. PERFECT ISOMETRIES IN TI BLOCKS 65 reason why we do not consider p = 2 here. Any reasonable bound on the size of D 2 ∼ 2 6 would include G = Aut( B2(8)) = B2(8) o C3 which has |D| = 2 .

Remark 3.6.7. Recall from Section 2.3.1 that two blocks B and b being derived equivalent, implies the existence of a perfect isometry from Irr(B) to Irr(b), which in turn induces an algebra isomorphism between Z(B) and Z(b). Hence if Z(B) 6∼= Z(b) then no such perfect isometry can exist. On the other hand, Brou´e’sabelian defect group conjecture (ADGC) states that if B has abelian defect groups then B and its Brauer correspondent b are derived equivalent and hence there exists a perfect isometry between the two blocks.

Theorem 3.6.8. Fix p = 3. Let G be a ﬁnite group and B0 ∈ Bl(G) be the principal 5 block with TI defect groups D such that |D| ≤ 3 ; let b0 ∈ Bl(NG(D)). Then there ∼ 1+2 exists a perfect isometry between B0 and b0 if and only if D is abelian or D = 3− .

Proof. We apply Lemma 3.6.3 and Remark 3.6.7, and individually consider the cases given in Lemma 3.6.4 which relate to p = 3. The structure descriptions of S and Aut(S) given below follow from [CCN+85].

2 2m+1 S = G2(3 ) 2 3 The smallest simple group is S = G2(3 ) and the defect group of the principal block has size |D| = 39. Hence this case is excluded in the statement.

S = PSL3(4) ∼ Suppose S ≤ G ≤ Aut(S). Then, by Lemma 3.6.5, p - [G : S] and D = C3 × C3. Hence the ADGC holds in this case [KK02].

2 0 S = G2(3) Note that |Out(S)| = 3 so G ∼= S or G ∼= Aut(S). If G ∼= S then D is cyclic and the ∼ 2 ADGC holds in this case ([Lin91], [Ric89], [Rou98]). If G = Aut(S) = G2(3), then ∼ 1+2 D = 3− and there exists a perfect isometry between B0 and b0 [HKK10, Example 4.3]; it is not known if the two blocks are derived equivalent.

S = M11 ∼ Since S has trivial outer automorphism group, G = S. The principal 3-block of M11 ∼ has abelian defect group D = C3 × C3, and the ADGC has been veriﬁed in this case [Oku98]. S = PSL(2, 3m) 66 CHAPTER 3. BLOCKS WITH TI DEFECT GROUPS

∼ m ∼ m If G = PSL2(3 ) where 1 ≤ m ≤ 5, then D = (C3) is abelian and the ADGC has been veriﬁed [Oku00]. If G is such that S < G ≤ Aut(S), then by Lemma 3.6.5, gcd(p, [G : S]) = 1. Hence by [Fon], there exists a perfect isometry between B0 and b0. S = PSU(3, 3m) ∼ ∼ If PSU(3, 3) = S ≤ G ≤ Aut(S) = PSU(3, 3) : C2, then D = (C3 o C3) o C3 is not ∼ abelian and by Theorem 3.4.1, Z(B0) 6= Z(b0); hence no perfect isometry can exist. If S = PSU(3, 3m) for m > 1, then |D| > 35.

We next consider Conjecture 3.6.6 for principal 5-blocks. The group PSU(3, 25) has a principal block with defect group D such that |D| = 56. Hence we restrict to blocks with defect groups of smaller sizes.

Theorem 3.6.9. Fix p = 5. Let G be a ﬁnite group and B0 ∈ Bl(G) be the principal 5 block with TI defect groups D such that |D| ≤ 5 ; let b0 ∈ Bl(NG(D)). Then there ∼ 1+2 exists a perfect isometry between B0 and b0 if and only if D is abelian or D = 5− .

Proof. As in Theorem 3.6.8, we apply Lemma 3.6.3 and Remark 3.6.7, and individually consider the cases given in Lemma 3.6.4 which relate to p = 5. The structure descriptions of S and Aut(S) given below follow from [CCN+85].

2 5 S = B2(2 ) Note that |Out(S)| = 5 so G ∼= S or G ∼= Aut(S). If G ∼= S then D is cyclic and the ∼ ∼ 1+2 ADGC holds in this case ([Lin91], [Ric89], [Rou98]). If G = Aut(S), then D = 5− and there exists a perfect isometry between B0 and b0 [HKK10, Example 4.4]; it is not known if the two blocks are derived equivalent.

2 0 S = F4(2) ∼ 2 0 ∼ 2 Note that |Out(S)| = 2 so G = S = F4(2) or G = Aut(S) = F4(2). In either case, ∼ D = C5 × C5 is abelian and the ADGC has been veriﬁed by Robbins [Rob08]. S = McL

Let S ≤ G ≤ Aut(S). Then the defect groups of B0(G) are not abelian and by ∼ Theorem 3.2.1 and Theorem 3.2.2, Z(B0) 6= Z(b0); hence no perfect isometry can exist. S = PSL(2, 5m) ∼ m ∼ m If G = PSL2(5 ) where 1 ≤ m ≤ 5, then D = (C5) is abelian and the ADGC has 3.6. PERFECT ISOMETRIES IN TI BLOCKS 67 been veriﬁed [Oku00]. If G is such that S < G ≤ Aut(S) then, by Lemma 3.6.5, gcd(p, [G : S]) = 1. Hence by [Fon], there exists a perfect isometry between B0 and b0. S = PSU(3, 5m) ∼ ∼ If PSU(3, 5) = S ≤ G ≤ Aut(S) = PSU(3, 5) : S3, then D = (C5 o C5) o C5 is not ∼ abelian and by Theorem 3.4.2, Z(B0) 6= Z(b0); hence no perfect isometry can exist. If S = PSU(3, 5m) for m > 1, then |D| > 55.

Remark. The two exceptions in Conjecture 3.6.6 arise from a weak conjecture of Brou´eand Rouquier; this is discussed in [HKK10, Conjectures 4.1] and we restate it here to give a more complete picture of the context of our results. Let B0 be the principal p-block of a ﬁnite group G with a non-abelian Sylow p-subgroup P . Let Q be the hyperfocal subgroup of P in G, Q = P ∩ H where H is the smallest normal subgroup of G satisfying that G/H is p-nilpotent. Rouquier conjectures that if Q is abelian then the p-block B0 and its Brauer correspondent b0 in NG(Q) should be derived equivalent; the weaker version, as stated by Koshitani, Holloway and Kunugi, conjectures the existence of a perfect isometry in this case. In most of our cases of non-abelian, trivial intersection defect groups, we have H = G and so Q = P ∩ H = P ; therefore Q is not abelian, and the weaker conjecture

2 0 does not apply. The only exceptions are precisely the examples G = Aut( G2(3) ) and 2 5 G = Aut( B2(2 )), as discussed in Examples 4.3 and 4.4 of [HKK10]. Chapter 4

On the Loewy length of the Suzuki groups and the small Ree groups in deﬁning characteristic

This chapter concerns two families of ﬁnite simple groups of Lie type: the Suzuki

2 2m+1 groups, denoted by B2(q) for q = 2 ≥ 8 [Suz60], and the Ree groups, denoted 2 2k+1 by G2(q) for q = 3 ≥ 27 [Ree61], where m, k ∈ N. The Sylow p-subgroups are trivial intersection for p = 2 and 3 respectively [GL83, Theorem 24.1]. However we ﬁrst establish a more general result, which holds for all ﬁnite groups with trivial intersection Sylow p-subgroups.

4.1 Relating structure constants

Proposition 4.1.1. Let G be a ﬁnite group with trivial intersection Sylow p-subgroups

g and Q ∈ Sylp(G). If x ∈ Q, g ∈ G such that x ∈ Q then either x = 1 or g ∈ NG(Q).

In particular, NG(Q) controls fusion of Q in G.

Proof. Assume x, xg ∈ Q, then   {1}, g 6∈ NG(Q); xg ∈ Q ∩ Qg =  Q, g ∈ NG(Q).

g g Thus if x 6= 1 then x 6= 1. Hence x ∈ Q and g ∈ NG(Q).

68 4.1. RELATING STRUCTURE CONSTANTS 69

Corollary 4.1.2. Let G be a ﬁnite group with trivial intersection Sylow p-subgroups

P . Then CG(x) ≤ NG(P ) for all x ∈ P .

The following proposition illustrates that for elements in a Sylow p-subgroup Q of G, the structure constant of the conjugacy class sums when multiplied as elements in

G is the same (mod p) as in NG(Q). The argument Cliﬀ gave for the Suzuki groups [Cli00, Lemma 3.2] in fact only requires that the Sylow p-subgroups have the trivial intersection property, hence yielding the following generalised result.

Proposition 4.1.3. Let G be a ﬁnite group with trivial intersection Sylow p-subgroups.

Let x, y, z ∈ Q \{1G} where Q ∈ Sylp(G). Then a(x, y, z) ≡ aH (x, y, z) mod |CG(z)|p where H = NG(Q).

Proof. Let

A = {(x0, y0) | x0 ∈ xG, y0 ∈ yG, x0y0 = z}.

Then |A| = a(x, y, z) and A can be split into two disjoint sets A1 ∪ A2, where

0 0 0 G 0 G 0 0 A1 = {(x , y ) | x ∈ x ∩ Q, y ∈ y , x y = z} 0 0 0 G 0 G 0 0 A2 = {(x , y ) | x ∈ x \ Q, y ∈ y , x y = z}

Note that as z = x0y0 ∈ Q, we have x0 ∈ xG ∩ Q if and only if y0 ∈ yG ∩ Q.

Claim 1 xG ∩ Q = xNG(Q) Suppose x0 ∈ xG ∩ Q. Then there exists an element g ∈ G such that xg = x0 ∈ Q∩xG. Since the Sylow p-subgroups of G are trivial intersection, by Proposition 4.1.1,

g h 0 g h NG(Q) there exists h ∈ NG(Q) such that x = x . Hence x = x = x ∈ x .

Therefore |A1| = aH (x, y, z).

Claim 2 The size of A2 is divisible by |CG(z)|p.

g Take an element z in Q \{1}. Suppose g ∈ NG(hzi), so hzi = hzi; this implies g g hzi ∈ Q ∩ Q = {1}, a contradiction unless g ∈ NG(Q). Hence the normaliser and thus the centraliser of z is a subgroup of NG(Q), i.e. CG(z) ≤ NG(hzi) ≤ NG(Q).

g 0 0 g 0g 0g Note that if g ∈ CG(z) then z = z = (x y ) = x y ; since CG(z) ≤ NG(Q), if 0 g 0 g (x ) ∈ Q then x ∈ Q = Q. Therefore CG(z) acts on A2. 70 CHAPTER 4. SUZUKI AND REE GROUPS

0 0 Suppose (x , y ) ∈ A2. By the orbit stabiliser theorem, the size of a CG(z)-orbit containing (x0, y0) is given by

0 0 CG(z) 0 0 |(x , y ) | = [CG(z):(CCG(z)(x ) ∩ CCG(z)(y ))] 0 0 = [CG(z):(CG(z) ∩ CG(x ) ∩ CG(z) ∩ CG(y ))]

0 0 g Consider CG(z) ∩ CG(x ), and suppose x = x such that g 6∈ NG(Q); then CG(z) ∩ g g g CG(x ) ≤ NG(Q) ∩ NG(Q ). Suppose S ∈ Sylp(NG(Q) ∩ NG(Q )). As S ≤ NG(Q) g g it follows that S ≤ Q; similarly S ≤ NG(Q ) implies S ≤ Q . Hence as g 6∈ NG(Q), g 0 S ≤ Q ∩ Q = 1 \{1}, and it follows that p does not divide |CG(z) ∩ CG(x )|. In 0 0 particular, p does not divide |CG(z) ∩ CG(x ) ∩ CG(y )|.

0 0 CG(z) 0 0 Hence |(x , y ) |p = |CG(z)|p for all pairs (x , y ) ∈ A2 and therefore |CG(z)|p divides the size of A2.

Finally, combining the two claims, |A| ≡ |A1| mod |CG(z)|p, and since A1 and A2 are disjoint, |A| ≡ |A1| ≡ aH (x, y, z) mod |CG(z)|p.

4.2 Suzuki Groups

∼ 2 2m+1 This section concerns the Suzuki groups, G = B2(q), q = 2 ≥ 8, m ∈ N. The main result of this section is that the centre of the principal block, Z(kGeo), and the centre of its Brauer correspondent Z(b0(kNG(P ))) both have Loewy length 2, where k is a ﬁeld of characteristic 2; hence they are isomorphic over k. This result was proven by Cliﬀ [Cli00, Theorem 4.1]; we will outline the proof in this section for completeness.

4.2.1 Introduction

Throughout this section, G denotes the Suzuki group of order q2(q −1)(q2 +1) [Suz60], where q = 22m+1 ≥ 8, m ∈ N. Our argument is based on that given by Cliﬀ [Cli00], however we simplify the argument by only requiring the value of the structure constants inside kG. In order to describe the conjugacy classes, we will use the following setup. There are cyclic subgroups A0,A1 and A2 of G, of orders q − 1, q + r + 1 and q − r + 1 √ respectively [Suz60], where r = 2q. In Table 4.2.1, πi denotes a representative of the conjugacy classes of non-identity elements of Ai. The element σ is an element of 4.2. SUZUKI GROUPS 71 the unique conjugacy class of involutions. Finally, there are two classes of elements of order 4: C (ρ) and C (ρ−1). The Sylow 2-subgroups of G intersect trivially [GL83, Theorem 24.1] . The generic character table for the Suzuki groups is given by Table 4.2.1; here √ q = 22m+1 ≥ 8 and r = 2q > 2.

2 Table 4.2.1: Character Table of B2(q) [Suz62]

number of −1 characters 1G σ ρ, ρ π0 π1 π2 1 X q2 0 0 1 −1 −1 2 i q/2 − 1 Xi q + 1 1 1 ε0(π0) 0 0 j (q + r)/4 Yj (q − r + 1)(q − 1) r − 1 −1 0 −ε1(π1) 0 (q − r)/4 k Zk (q + r + 1)(q − 1) −r − 1 √−1 0 0 −ε2(π2) 2 Wl r(q − 1)/2 −r/2 ±r −1/2 0 1 −1

2 Table 4.2.2: Class sizes and centraliser orders in B2(q) [Suz62]

−1 g σ ρ, ρ π0 π1 π2 |C (g)| (q2 + 1)(q − 1) q(q2 + 1)(q − 1)/2 q2(q2 + 1) q2(q − 1)(q − r + 1) q2(q − 1)(q + r + 1) 2 |CG(g)| q 2q q − 1 q + r + 1 q − r + 1

In characteristic 2, the group algebra kG decomposes as the direct sum of the principal block B0 and one block of defect zero, containing the Steinberg character X [BM90, Proposition 6.3]. The corresponding block idempotents are given by

X(1) P −1 eˆX = |G| X(g )g g∈G 1 2 P P P = (q−1)(q2+1) q + Cb(π0) − Cb(π1) − Cb(π2) π0 π1 π2

P P P ≡ Cb(π0) + Cb(π1) + Cb(π2) mod J(O)G π0 π1 π2 and e0 = 1 − eX ∈ kG.

4.2.2 Main Theorem for Suzuki Groups

Throughout this subsection, σ, ρ and πi are elements as given in Table 4.2.1. 72 CHAPTER 4. SUZUKI AND REE GROUPS

2 2m+1 Theorem 4.2.1. [Cli00, Theorem 4.1] Let G = B2(q) where q = 2 ≥ 8, and k is an algebraically closed ﬁeld of characteristic 2. Then LL(Z(kGe0)) = 2.

Let DG = {Cb(g)e0 | g ∈ P, g 6= 1G, g 6∈ C (σ)} ∪ {(1 + Cb(σ))e0}. Then spank(DG) is k-linear spanning set for J(Z(kGe0)) and spank(DG ∪ {e0}) for Z(kGe0).

Recall the following notation for the multiplication of two class sums:

X Cb(x)Cb(y) = a(x, y, z)Cb(z), x, y ∈ G. z∈P

We use Burnside’s formula, Formula 1.2.2, to calculate a(x, y, z).

Proposition 4.2.2. Let x, y ∈ G \{1G}. Then

 −1  0, if x ∈ C (ρ) or C (ρ ), y ∈ C (πi); Cb(x) · Cb(y) =  eX , if x, y ∈ C (πi).

Proof. We will split this proof into two cases, corresponding to the diﬀerent behaviours of the conjugacy classes. All equivalences are taken modulo J(O)G.

−1 Case 1 Suppose x ∈ C (ρ) or C (ρ ) and y ∈ C (πi). By Table 4.2.2, |CG(x)| = 2q and |CG(y)| = s where s is an odd integer. Hence

q2(q−1)(q2+1) P χ(x)χ(y)χ(z−1) a(x, y, z) = 2q·s χ(1) where gcd(q, s) = 1 χ∈Irr(G)

−1 √ −1 2m P χ(x)χ(y)χ(z ) P (±r −1/2)(±1)Wl(z ) ≡ 2 χ(1) + r(q−1)/2 χ∈Irr(G)\{Wl,X} Wl∈Irr(G)

X(ρ±1)X(y)X(z−1) + X(1) ! −1 −1 2m P χ(x)χ(y)χ(z ) P (±i)(±1)Wl(z ) = 2 χ(1) + q−1 + 0 χ∈Irr(G)\{Wl,X} Wl∈Irr(G)

≡ 22m ≡ 0 mod J(O)G

For the fourth equivalence, we make use of the fact the for χ ∈ Irr(G) \{Wl,X}, 2 does not divide χ(1G).

Case 2 Suppose x, y ∈ C (πi). By Table 4.2.2, the centraliser orders are |CG(x)|, 4.2. SUZUKI GROUPS 73

|CG(y)| ∈ {q − 1, q + r + 1, q − r + 1}, an odd integer. Hence

2 2 −1 a(x, y, z) = q (q−1)(q +1) P χ(x)χ(y)χ(z ) |CG(x)|·|CG(y)| χ(1) χ∈Irr(G) ! 2 X(x)X(y)X(z−1) P χ(x)χ(y)χ(z−1) ≡ q q2 + χ(1) χ∈Irr(G)\{X}

2 X(x)X(y)X(z−1) ≡ q q2 + 0

2 ±1·X(z−1) ≡ q q2

≡ X(z−1)  −1  0, C (z) ∈ {C (1G), C (σ), C (ρ), C (ρ )}; ≡  1, z ∈ C (πi).

Therefore, in this case, Cb(x)Cb(y) = eX .

Proposition 4.2.3. Let x ∈ C (σ), and C (y) ∈ S = {C (π0), C (π1), C (π2)}. Then

Cb(x) · Cb(y) = eX − Cb(y).

Proof. We note that S consists of (q + 3) − 4 = q − 1 conjugacy classes. For C (y) in S we have so far calculated all structure constants a(x, y, z) (see Proposition 4.2.2), except a(σ, y, z). Hence we can use Proposition 1.2.7 to ﬁnd the remaining one. P We have |C (y)| ≡ 0 mod 2. Hence |C (y)| = x a(x, y, z) ≡ 0. Let α = a(σ, y, z). Then

X −1 a(x, y, z) = a(1G, y, z) + a(ρ, y, z) + a(ρ , y, z) + (q − 1)a(x ∈ S, y, z) + α x

  1 + 0 + 0 + (q − 1)(+1) + α ≡ α, if C (y) = C (z) ∈ S;  = 0 + 0 + 0 + (q − 1)(+1) + α ≡ −1 + α if C (y) 6= C (z) and C (z) ∈ S;   0 + 0 + 0 + (q − 1)(0) + α ≡ α if C (z) 6∈ S.   0, if C (y) = C (z) ∈ S;  ⇒ a(σ, y, z) = 1, if C (y) 6= C (z) and C (z) ∈ S;   0, if C (z) 6∈ S. 74 CHAPTER 4. SUZUKI AND REE GROUPS

Proposition 4.2.4. [Cli00, Proposition 3.1] Let x, y ∈ G \{1G}. Then  −1  0, if x, y ∈ C (ρ) or C (ρ );  Cb(x) · Cb(y) = Cb(x), if x ∈ C (ρ) or C (ρ−1), y ∈ C (σ);    e0, if x, y ∈ C (σ).

Proof. We can apply Proposition 4.1.3. Hence for x, y, z ∈ P \{1G} where P ∈ Syl2(G) and H = NG(P ), we have a(x, y, z) ≡ aH (x, y, z) ≡ 0 mod |CG(z)|2 by Theorem 4.2.5. Furthermore, for each of the three cases, use the tables on pages 85, 84 and 83 of [Cli00], respectively.

Proof of Theorem 4.2.1. We simply go through all the possible multiplications of two elements in DG. Note that Cb(x), Cb(y), e0 are all in Z(kG), hence commute, and 2 e0 = e0. We have the following equivalences modulo J(O)G:

2 2 • (1 + Cb(σ)) e0 ≡ (1 + Cb(σ) )e0 = (1 + e0)e0 = e0 + e0 ≡ 0;

±1 ±1 ±1 • Cb(ρ )(1 + Cb(σ))e0 ≡ (Cb(ρ ) + Cb(ρ ))e0 ≡ 0;

±1 ±1 • Cb(ρ )Cb(ρ )e0 ≡ 0.

±1 • Cb(ρ )Cb(πi)e0 ≡ 0.

• Cb(πi)Cb(πj)e0 ≡ eX e0 = 0.

• Cb(πi)(1 + Cb(σ))e0 ≡ (Cb(πi) + (eX − Cb(πi)))e0 ≡ eX e0 = 0.

4.2.3 Normaliser of Sylow 2-subgroups

Cliﬀ also proves the following result.

2 Theorem 4.2.5. [Cli00, Proposition 2.1] Let N = NG(P ) where G = B2(q), q = 2m+1 2 ≥ 8, and P ∈ Syl2(G). Then LL(Z(kN)) = 2.

Hence combining Theorem 4.2.1 and Theorem 4.2.5, it follows that there exists an isomorphism between the two centres over k. 4.2. SUZUKI GROUPS 75

2 Theorem 4.2.6. [Cli00, Theorem 4.1] Let G = B2(q), N = NG(P ) and k an algebraically closed ﬁeld of characteristic p = 2. Then LL(Z(kGe0)) = LL(Z(kN)) = 2. ∼ Moreover, Z(kGe0) = Z(kN).

In addition, Theorem 4.1 of [Cli00] establishes that the centres are not isomorphic over ∼ a discrete valuation ring O, of characteristic 0, i.e. Z(OGe0) 6= Z(ON). Hence there is no perfect isometry between the principal block of the Suzuki groups and its Brauer correspondent (also see [Rob00] for an independent proof of this result); moreover no derived equivalence can exist.

2 2 Remark. We note that Gramain has proved that for G = B2(q) or G2(q), there 2 is a generalised perfect isometry with respect to p -regular elements between B0(kG) and kNG(P ) [Gra05, Theorem 2.6]. However, as noted in Section 2.2 of [Gra05], the existence of a generalised perfect isometry does not imply an algebra isomorphism between Z(B0(kG)) and Z(kNG(P )). 76 CHAPTER 4. SUZUKI AND REE GROUPS 4.3 The Ree groups

For the rest of this chapter, we turn our attention to another inﬁnite family of simple groups of Lie type, namely the small Ree groups. The main result, presented in

Theorem 4.3.23, states that the centre of the principal block Z(kGeo) has Loewy length 3, while the centre of its Brauer correspondent in kNG(P ) only has Loewy length 2, where k is an algebraically closed ﬁeld of characteristic 3. Hence the two centres cannot be isomorphic.

4.3.1 Introduction

2 3 3 Throughout this section, G will denote the small Ree group G2(q) of order q (q + 1)(q − 1) [Ree61], with q = 32k+1 ≥ 27. The argument follows the approach for the Suzuki groups described in Section 4.2, with some adjustments to the statements as necessary. The groups G were first described by Ree [Ree61] and their structure was determined in detail by Ward [War66], including their character table. Our notation for the conjugacy classes and characters follows the notation introduced in [War66]. For k an algebraically closed field of characteristic 3, the group algebra kG decomposes into two blocks: the principal 3-block B0(kG) and one block of defect zero 3 containing the Steinberg character ξ3 of degree q (see Table 4.3.2). The outer automorphism group is cyclic, generated by the field automorphism τ of GF (q) with order

2k + 1. Furthermore, NG(P ) has one block, b, which is the Brauer correspondent of

B0(kG). The group G has q +8 conjugacy classes [War66, p. 85], k(B0) = q +7 = k(b) and l(B0) = q − 1 = l(b) [BM90, Proposition 6.2]. Hence dim(Z(kGe0)) = q + 7. Let P be a Sylow 3-subgroup of G. Then |P | = q3, its centre Z(P ) is elementary abelian of order q, and the normaliser is given by NG(P ) = P oW where W is cyclic of order q − 1. The Sylow 3-subgroups have the trivial intersection property; this follows from Theorem (24.1) of Gorenstein-Lyons [GL83, p.307].

The group G has cyclic subgroups A0, A1, A2 and A3 [Jon94, Section 3], which have the following odd orders

1 1 |A | = (q − 1), |A | = (q + 1), |A | = q − 3m + 1, |A | = q + 3m + 1, 0 2 1 4 2 3 where m = 3k. Note that (q3 +1) = (q+1)(q2 −q+1) = (q+1)(q−3m+1)(q+3m+1). 4.3. THE REE GROUPS 77

a a The non-identity elements g ∈ Ai are denoted by R ,S ,Vi,Wi for which |CG(g)| = q − 1, q + 1, q − 3m + 1, q + 3m + 1 respectively, and |NG(Ai): Ai| = 4, 24, 6, 6 for i = 0, 1, 2, 3. The Sylow 2-subgroups of G are elementary abelian of order 8. There is a single conjugacy class of involutions, represented by J, with centraliser isomorphic to C2 ×

PSL2(q) [War66, Chapter 1]. 2 Most values of the character table for the Ree groups G2(q) were determined by Ward [War66]. We explicitly write down the information contained in the paper, which is needed in the calculations to follow. The notation is consistent with Ward’s, i.e. q = 32k+1 and m = 3k.

Table 4.3.1: Conjugacy classes, their centraliser orders and corresponding defects

conjugacy class order |CG(g)| dg 1G 1 |G| 3(2m + 1) Ra, 1 ≤ a ≤ (q − 3)/4 (q − 1)/2 q − 1 0 a S , 1 ≤ a ≤ (q − 3)/24 (q √+ 1)/4 q + 1 0 Vi, 1 ≤ i ≤ (q − 3m)/6 q − √3q + 1 q + 1 − 3m 0 Wi, 1 ≤ i ≤ (q − 3m)/6 q + 3q + 1 q + 1 + 3m 0 X 3 q3 3(2m + 1) Y 9 3q (2m + 1) + 1 T 3 2q2 2(2m + 1) T −1 3 2q2 2(2m + 1) YT 9 3q (2m + 1) + 1 YT −1 9 3q (2m + 1) + 1 JT 6 2q (2m + 1) JT −1 6 2q (2m + 1) JRa, 1 ≤ a ≤ (q − 3)/4 2|Ra| q − 1 0 JSa, 1 ≤ a ≤ (q − 3)/8 2|Sa| q + 1 0 J 2 (q + 1)q(q − 1) (2m + 1)

Note that for an element g in one of the six families of conjugacy classes, Ra, Sa,

a a Vi, Wi, JR , JS , the elements don’t have the order stated, but rather their order divides the given value. Let

a a a a S = {R ,S ,Vi,Wi,JR ,JS };

|S| = q − 2. With slight abuse of notation, we also say that a conjugacy class C (g) is in S or a conjugacy class sum Cb(g) ∈ S, if we want to refer to one of the six families of conjugacy classes. 78 CHAPTER 4. SUZUKI AND REE GROUPS

Table 4.3.2: The irreducible characters of G [War66]

number of number of χ χ(1G) characters χ χ(1G) characters 2 ξ1 1 1 ξ9 m(q − 1) 1 2 2 ξ2 q − q + 1 1 ξ10 m(q − 1) 1 3 3 ξ3 q 1 ηr q + 1 (q − 3)/4 2 0 3 ξ4 q(q − q + 1) 1 ηr q + 1 (q − 3)/4 2 ξ5 (q − 1)m(q + 1 + 3m)/2 1 ηt (q − 1)(q − q + 1) (q − 3)/24 0 2 ξ6 (q − 1)m(q + 1 − 3m)/2 1 ηt (q − 1)(q − q + 1) (q − 3)/8 − 2 ξ7 (q − 1)m(q + 1 + 3m)/2 1 ηi (q − 1)(q + 1 + 3m)(q − 3m)/6 + 2 ξ8 (q − 1)m(q + 1 − 3m)/2 1 ηi (q − 1)(q + 1 − 3m)(q + 3m)/6

As mentioned above, Ward’s character table [War66] is not quite complete. Some entries are missing; however since only their sum is of interest to us, orthogonality of the columns can be applied to ﬁnd the required information. The following table summaries which columns are used for the orthogonality relations, and which results can be obtained from this.

Columns Results

a P a P 0 a R ,JT r ηr(R ) = r ηr(R )

a P a R , 1 r ηr(R ) = −1

a P a P 0 a S ,Y 4 − t ηt(S ) − t ηt(S ) = 0

a P a P 0 a S ,J t ηt(S ) = 1, t ηt(S ) = 3

P − Vi,Y i ηi (Vi) = 1

P + Wi,Y i ηi (Wi) = 1

a P a P 0 a JR , 1 r ηr(JR ) = − r ηr(JR )

a P a P 0 a JR ,JT r ηr(JR ) = −1, r ηr(JR ) = 1

a P a P 0 a JS ,Y − t ηt(JS ) − t ηt(JS ) = 0

a P a P 0 a JS ,J t ηt(JS ) = 1, t ηt(S ) = −1

We now write down the two block idempotents, where the equivalences are taken 4.3. THE REE GROUPS 79 modulo J(O)G.

ξ3(1) P −1 eˆξ3 = |G| ξ3(g )g g∈G (q−3)/4 (q−3)/24 (q−3m)/6 q3 3 P a P a P = q3(q3+1)(q−1) q + Cb(R ) − Cb(S ) − Cb(Vi) a=1 a=1 i=1 (q+3m)/6 (q−3)/4 (q−3)/8 ! P P a P a − Cb(Wi) + Cb(JR ) − Cb(JS ) + q · Cb(J) i=1 a=1 a=1 (q−3)/4 (q−3)/24 (q−3m)/6 (q+3m)/6 P a P a P P ≡ −1 · Cb(R ) − Cb(S ) − Cb(Vi) − Cb(Wi) a=1 a=1 i=1 i=1 (q−3)/4 (q−3)/8 ! + P Cb(JRa) − P Cb(JSa) a=1 a=1 (q−3)/4 (q−3)/24 (q−3m)/6 (q+3m)/6 P a P a P P ≡ − Cb(R ) + Cb(S ) + Cb(Vi) + Cb(Wi) a=1 a=1 i=1 i=1 (q−3)/4 (q−3)/8 − P Cb(JRa) + P Cb(JSa) a=1 a=1

and e0 = 1 − eξ3 ∈ kG.

2 4.3.2 The Example G2(27)

2 Let G = G2(27); this is the smallest simple Ree group, and we can calculate the structure constants in [GAP] using the code given in Appendix A. This will indicate the expected behaviour in a generic Ree group. Moreover, in the proof of Lemma 4.3.6, we have to assume that q > 27; hence we prove Theorem 4.3.1 for q = 27 here. The group algebra kG decomposes into two blocks: the principal block of defect 9 and one block of defect zero corresponding to the character ξ3 in Table 4.3.2. Note that 2 + ξ3 is denoted by χ20 in the character table of G2(27) as given in the Atlas [CCN 85].

The block idempotent for the principal block B0(kG) is given by

e0 = 1 − eχ20 6 3 3 6 6 P P P P P = 1 − Cb(7) − Cb(13i) + Cb(14i) + Cb(19i) − Cb(26i) + Cb(37i) . i=1 i=1 i=1 i=1 i=1

From Section 4.3.1, dim(Z(kGe0)) = q + 7 = 34 = dim(Z(kNG(P ))). Take

DG = {Cb(x)e0 | x ∈ P, x 6= 1G, x 6∈ C (3A)} ∪ {(Cb(3A) + 1)e0}

as the spanning set for J(Z(kGe0)). The following table summarises all the possible multiplications of two conjugacy class sums in DG, ignoring the block idempotent e0. 80 CHAPTER 4. SUZUKI AND REE GROUPS

2A 1 + 3A 3B 3C 6A,B 7A 9A,B,C 13A,...,F 14A,B,C 19A,B,C 26A,...,F 37A,...,F

2A eξ3 eξ3 − − − eξ3 − eξ3 eξ3 eξ3 eξ3 eξ3

1 + 3A α β β − eξ3 − eξ3 eξ3 eξ3 eξ3 eξ3

3B β β − − − − − − − −

3C β − − − − − − − −

6A,B − − − − − − − −

7A eξ3 − eξ3 eξ3 eξ3 eξ3 eξ3

9A,B,C − − − − − −

13A,...,F eξ3 eξ3 eξ3 eξ3 eξ3

14A,B,C eξ3 eξ3 eξ3 eξ3

19A,B,C eξ3 eξ3 eξ3

26A,...,F eξ3 eξ3

37A,...,F eξ3

Here “ − ” denotes a zero in kG and

6 3 6 P P P α = Cb(7A) + Cb(13i) + Cb(19i) + Cb(37i); i=1 i=1 i=1 6 3 6 P P P β = 2 Cb(13i) + 2 Cb(14i) + Cb(26i). i=1 i=1 i=1

Since eξ3 ·e0 = 0, we can see from the table that most pairs of elements in DG multiply to zero. The only non-zero multiplications are:

6 3 6 2 P P P (1 + Cb(3A)) e0 = Cb(7A) + Cb(13i) + Cb(19i) + Cb(37i) e0 i=1 i=1 i=1 6 3 6 P P P = 2 Cb(13i) + 2 Cb(14i) + Cb(26i) i=1 i=1 i=1

2 2 (1 + Cb(3A))Cb(3B)e0 = (1 + Cb(3A))Cb(3C)e0 = Cb(3B) e0 = Cb(3C) e0 6 3 6 P P P = Cb(3B) · Cb(3C)e0 = 2 Cb(13i) + 2 Cb(14i) + Cb(26i) e0 i=1 i=1 i=1 6 3 6 P P P = 2 Cb(13i) + 2 Cb(14i) + Cb(26i) i=1 i=1 i=1

Note that none of the outcomes of these multiplications involve the conjugacy classes C (3A), C (3B) or C (3C). Hence any triple will multiply to zero and therefore

LL(Z(kGe0)) = 3. 4.3. THE REE GROUPS 81

2 Next consider the normaliser of a Sylow 3-subgroup of G2(27), denoted N. The group algebra kN is indecomposable, and we take the following set as a basis for J(Z(kN)) (see Proposition 1.5.10):

BN = {Cb(x) | x ∈ P, x 6= 1N , x 6∈ C (3A)} ∪ {Cb(3A) + 1}.

0 0 In GAP we calculate that for all b, b ∈ BN , b · b = 0 yielding LL(Z(kN)) = 2.

In particular, LL(Z(kGe0)) > LL(Z(kNG(P ))), so the centre of the principal block of kG cannot be isomorphic to the centre of its Brauer correspondent in kNG(P ).

4.3.3 Main Theorem for Ree groups

2 In the previous section, it was established that for G = G2(27), LL(Z(kGe0)) = 3. 2 The aim now is to show that this Loewy length is 3 for a generic Ree group G2(q) where q = 32k+1. In particular, over a series of lemmas comprising the whole of this section, the following theorem is proven.

2 2k+1 Theorem 4.3.1. Let G = G2(q) where q = 3 ≥ 27, and k an algebraically closed

ﬁeld of characteristic 3. Then LL(Z(kGe0)) = 3.

By Table 4.3.1, all non-trivial conjugacy classes have class size divisible by 3 except C (X) which has size |C (X)| = (q3 + 1)(q − 1). Hence by Remark 1.5.18, a spanning set for J(Z(kGe0)) is given by

DG = {Cb(x)e0 | x ∈ P, x 6= 1G, x 6∈ C (X)} ∪ {(Cb(X) + 1)e0}.

2 Following the argument presented for G2(27) in Section 4.3.2, ﬁrst the multiplication of two conjugacy class sums in DG is considered, ignoring the block idempotent e0. For a clear overview, the outcomes of these multiplications are summarised in Table 4.3.4. By using the notion of defect for a conjugacy class, some structure constants can immediately be seen to be zero.

Lemma 4.3.2. Let y ∈ S and let z be a 3-singular element or an involution. Then a(x, y, z) ≡ 0 mod J(O)G for all x ∈ G.

Proof. This is a direct application of Theorem 1.2.5 and the defects of the conjugacy classes as given in Table 4.3.1. 82 CHAPTER 4. SUZUKI AND REE GROUPS

Using Burnside’s formula

Lemma 4.3.3. Let Cb(x), Cb(y) ∈ S. Then Cb(x) · Cb(y) = eξ3 .

Proof. For the conjugacy classes C (x), C (y) as given in the statement, the centraliser orders |CG(x)| and |CG(y)| are not divisible by 3.

Table 4.3.3:

coeﬃcient of

element |CG(g)| mod 3 ξ3(g) Cb(g) in eξ3 Ra −1 +1 −1 Sa +1 −1 +1 Vi +1 −1 +1 Wi +1 −1 +1 JRa −1 +1 −1 JSa +1 −1 +1

The structure constants can now be calculated, where the equivalences are taken modulo J(O)G.

3 3 −1 a(x, y, z) = q (q−1)(q +1) P χ(x)χ(y)χ(z ) |CG(x)|·|CG(y)| χ(1) χ∈Irr(G) ! 3 −1 −1 3 (q−1)(q +1) ξ3(x)ξ3(y)ξ3(z ) P χ(x)χ(y)χ(z ) = q · 3 + |CG(x)|·|CG(y)| q χ(1) χ∈Irr(G)\{ξ3}

−1 3 −1 ξ3(x)ξ3(y)ξ3(z ) ≡ q · −1α · q3 + 0 where α ∈ {1, 2}

α −1 3 −1 −1 ·ξ3(z ) ≡ q · −1α · q3 where α ∈ {1, 2}

−1 ≡ −1 · ξ3(z )   0 mod 3, z 6∈ S; ≡  ±1 mod 3, z ∈ S.

Here the ±1 corresponds to the same coeﬃcient the class sum of C (z) has in the block idempotent eξ3 given in Table 4.3.3. This proves the claim.

Lemma 4.3.4. Let x ∈ C (JT ) or C (JT −1), and C (y) ∈ S. Then Cb(x) · Cb(y) = 0. 4.3. THE REE GROUPS 83

Proof. From Table 4.3.1, |CG(x)| = 2q and |CG(y)| ≡ ±1 mod 3.

q3(q−1)(q3+1) P χ(x)χ(y)χ(z−1) a(x, y, z) = 2q·s χ(1) where gcd(q, s) = 1 χ∈Irr(G) ! −1 −1 2 ξ3(x)ξ3(y)ξ3(z ) P χ(x)χ(y)χ(z ) ≡ q · s q3 + χ(1) ≡ 0 mod J(O)G χ∈Irr(G)\{ξ3}

−1 2 The last equivalence follows from the facts that ξ3(JT ) = ξ3(JT ) = 0 and q does not divide χ(1) for all χ ∈ Irr(G) \{ξ3}.

Lemma 4.3.5. Let x ∈ C (Y ), C (YT ) or C (YT −1), and C (y) ∈ S. Then Cb(x) · Cb(y) = 0.

Proof. From Table 4.3.1, |CG(x)| = 3q and |CG(y)| ≡ ±1 mod 3.

q3(q−1)(q3+1) P χ(x)χ(y)χ(z−1) a(x, y, z) = 3q·s χ(1) where gcd(q, s) = 1 χ∈Irr(G) ! 2 −1 −1 q ·s ξ3(x)ξ3(y)ξ3(z ) P χ(x)χ(y)χ(z ) ≡ 3 q3 + χ(1) ≡ 0 mod J(O)G χ∈Irr(G)\{ξ3}

−1 The last equivalence follows from the facts that ξ3(Y ) = ξ3(YT ) = ξ3(YT ) = 0 and 2 q /3 does not divide χ(1) for all χ ∈ Irr(G) \{ξ3}.

Lemma 4.3.6. Let x, y ∈ C (Y ), C (YT ) or C (YT −1). Then Cb(x) · Cb(y) = 0.

Proof. From Table 4.3.1, |CG(x)| = |CG(y)| = 3q.

a(x, y, z) q3(q−1)(q3+1) P χ(x)χ(y)χ(z−1) = 32q2 χ(1) χ∈Irr(G) ! −q P χ(x)χ(y)χ(z−1) P χ(x)χ(y)χ(z−1) P χ(x)χ(y)χ(z−1) ≡ 32 χ(1) + χ(1) + χ(1) χ∈Irr(G)\{ξ3,...,ξ10} χ∈{ξ3,ξ4} χ∈{ξ5,...,ξ10} ! 2k−1 P χ(x)χ(y)χ(z−1) P 0·0·χ(z−1) P χ(x)χ(y)χ(z−1) = −3 χ(1) + χ(1) + 3k·s χ∈Irr(G)\{ξ3,...,ξ10} χ∈{ξ3,ξ4} χ∈{ξ5,...,ξ10} where the equivalences are take modulo J(O)G and gcd(q, s) = 1. The last equivalence

−1 follows from the fact that ξ3,4(Y ) = ξ3,4(YT ) = ξ3,4(YT ) = 0. Now suppose q = 32k+1 > 27 (the result is true for q = 27 by the calculations in Section 4.3.2). Then

2k−1 k 3 > 3 , and since 3 does not divide χ(1) for all χ ∈ Irr(G) \{ξ3, . . . , ξ10}, we have a(x, y, z) ≡ 0 mod J(O)G. 84 CHAPTER 4. SUZUKI AND REE GROUPS

Lemma 4.3.7. Let x, y ∈ C (JT ) or C (JT −1). Then Cb(x) · Cb(y) = 0.

Proof. From Table 4.3.1, |CG(x)| = |CG(y)| = 2q.

q3(q−1)(q3+1) P χ(x)χ(y)χ(z−1) a(x, y, z) = 22q2 χ(1) χ∈Irr(G) ! P χ(x)χ(y)χ(z−1) P χ(x)χ(y)χ(z−1) ≡ −q · χ(1) + χ(1) χ∈Irr(G)\{ξ3,ξ4} χ∈{ξ3,ξ4} ! P χ(x)χ(y)χ(z−1) P 0·0·χ(z−1) = −q · χ(1) + χ(1) χ∈Irr(G)\{ξ3,ξ4} χ∈{ξ3,ξ4}

≡ 0. where the equivalences are take modulo J(O)G. The last equivalence follows from

−1 the fact that ξ3,4(JT ) = ξ3,4(JT ) = 0 and q does not divide χ(1) for all χ ∈

Irr(G) \{ξ3, ξ4}.

Lemma 4.3.8. Let x ∈ C (Y ), C (YT ) or C (YT −1) and y ∈ C (JT ) or C (JT −1). Then Cb(x) · Cb(y) = 0.

Proof. From Table 4.3.1, |CG(x)| = 3q and |CG(y)| = 2q.

a(x, y, z) q3(q−1)(q3+1) P χ(x)χ(y)χ(z−1) = 32q2 χ(1) χ∈Irr(G) ! −q P χ(x)χ(y)χ(z−1) P χ(x)χ(y)χ(z−1) P χ(x)χ(y)χ(z−1) ≡ 3·2 χ(1) + χ(1) + χ(1) χ∈Irr(G)\{ξ3,...,ξ10} χ∈{ξ3,ξ4} χ∈{ξ5,...,ξ10} ! 2k P χ(x)χ(y)χ(z−1) P 0·0·χ(z−1) P χ(x)χ(y)χ(z−1) = 3 χ(1) + χ(1) + 3k·s χ∈Irr(G)\{ξ3,...,ξ10} χ∈{ξ3,ξ4} χ∈{ξ5,...,ξ10} where the equivalence is taken modulo J(O)G and gcd(q, s) = 1. The last equality

−1 −1 follows from the fact that ξ3,4(Y ) = ξ3,4(YT ) = ξ3,4(YT ) = ξ3,4(JT ) = ξ3,4(JT ) = 0. Now by assumption of the group, k ≥ 1 and so 32k > 3k. Furthermore 3 does not divide χ(1) for all χ ∈ Irr(G) \{ξ3, . . . , ξ10}. Hence a(x, y, z) ≡ 0 modulo J(O)G.

Lemma 4.3.9. Let x ∈ C (J). Then   ξ3, if C (y) ∈ S or {C (J)}; Cb(x) · Cb(y) =  0, if C (y) ∈ {C (Y ), C (YT ), C (YT −1), C (JT ), C (JT −1)}. 4.3. THE REE GROUPS 85

Proof. We will split this proof into several cases. Note that |CG(J)| = (q + 1)q(q − 1).

Case 1: Suppose C (y) ∈ S. Then |CG(y)| ≡ ±1 mod 3 as given in Table 4.3.3.

3 2 −1 a(x, y, z) = q (q−1)(q+1)(q −q+1) P χ(x)χ(y)χ(z ) |CG(x)|·|CG(y)| χ(1) χ∈Irr(G) ! −1 −1 2 1 ξ3(x)ξ3(y)ξ3(z ) P χ(x)χ(y)χ(z ) = q · ±1 q3 + χ(1) χ∈Irr(G)\{ξ3}

−1 2 1 ξ3(x)ξ3(y)ξ3(z ) ≡ q · ±1 · q3

−1 2 1 q·∓1·ξ3(z ) ≡ q · ±1 · q3  0 mod 3, z 6∈ S; −1  ≡ −1 · ξ3(z ) ≡  ±1 mod 3, z ∈ S.

Here the ±1 corresponds to the same coeﬃcient the class sum of C (z) has in the block idempotent eξ3 (see Table 4.3.3). Hence in this case Cb(x) · Cb(y) ≡ eξ3 mod J(O)G. Case 2: Suppose C (y) = C (J).

3 2 −1 a(x, y, z) = q (q−1)(q+1)(q −q+1) P χ(x)χ(y)χ(z ) |CG(x)|·|CG(y)| χ(1) χ∈Irr(G) ! −1 −1 1 ξ3(x)ξ3(y)ξ3(z ) P χ(x)χ(y)χ(z ) = q · −1 q3 + χ(1) χ∈Irr(G)\{ξ3}

−1 1 ξ3(x)ξ3(y)ξ3(z ) ≡ q · −1 · q3

2 −1 1 q ·ξ3(z ) ≡ q · −1 · q3  0 mod 3, z 6∈ S; −1  ≡ −1 · ξ3(z ) ≡  ±1 mod 3, z ∈ S.

Here the ±1 corresponds to the same coeﬃcient the class sum of C (z) has in the block idempotent eξ3 (see Table 4.3.3). Hence in this case Cb(x) · Cb(y) ≡ eξ3 mod J(O)G. −1 Case 3 Suppose y ∈ C (Y ), C (YT ) or C (YT ). Then |CG(y)| = 3q, and

q3(q−1)(q+1)(q2−q+1) P χ(x)χ(y)χ(z−1) a(x, y, z) = q(q+1)(q−1)·3q χ(1) χ∈Irr(G) ! 2k P χ(x)χ(y)χ(z−1) P q·0·χ(z−1) P χ(x)χ(y)χ(z−1) ≡ 3 χ(1) + χ(1) + 3k·s χ∈Irr(G)\{ξ3,...,ξ10} χ∈{ξ3,ξ4} χ∈{ξ5,...,ξ10} 86 CHAPTER 4. SUZUKI AND REE GROUPS where gcd(q, s) = 1 and the equivalence is taken modulo J(O)G. The equivalence

−1 follows from the fact that ξ3,4(Y ) = ξ3,4(YT ) = ξ3,4(YT ) = 0. Now by assumption of the group, k ≥ 1 and so 32k > 3k. Furthermore 3 does not divide χ(1) for all

χ ∈ Irr(G) \{ξ3, . . . , ξ10}; hence a(x, y, z) ≡ 0 mod J(O)G. −1 Case 4 Suppose y ∈ C (JT ) or C (JT ). Then |CG(y)| = 2q, and

q3(q−1)(q+1)(q2−q+1) P χ(x)χ(y)χ(z−1) a(x, y, z) = (q+1)q(q−1)·2q χ(1) χ∈Irr(G) ! P χ(x)χ(y)χ(z−1) P χ(x)χ(y)χ(z−1) ≡ −q · χ(1) + χ(1) χ∈Irr(G)\{ξ3,ξ4} χ∈{ξ3,ξ4} ! P χ(x)χ(y)χ(z−1) P ±q·0·χ(z−1) = −q χ(1) + χ(1) ≡ 0 χ∈Irr(G)\{ξ3,ξ4} χ∈{ξ3,ξ4}

−1 The last equivalence follows from the fact that ξ3,4(JT ) = ξ3,4(JT ) = 0 and q does not divide χ(1) for all χ ∈ Irr(G) \{ξ3, ξ4}. This concludes the proof of this lemma.

Lemma 4.3.10. Let x ∈ C (J) and y ∈ C (T ) or C (T −1). Then Cb(x) · Cb(y) = 0.

Proof. By applying Theorem 1.2.5,

a(x, y, z) ≡ 0 mod J(O)G if z ∈ {X,Y,T,T −1,YT,YT −1}; while by [Jon94, Lemma 4.1],

a a a(x, y, z) ≡ 0 mod J(O)G if z ∈ {R ,S ,Vi,Wi}.

Note that the formula used in the calculations by Jones [Jon94], and hence the structure constants calculated, diﬀers up to a scalar; we can adjust appropriately by dividing by |C (z)|.

This leaves a(x, y, z) for z ∈ JT,JT −1,JRa,JSa which can be calculated directly from the character table. In particular

a(J, T, JT ) = 9 · m4 + 3 · m2; a(J, T, JT −1) = 0; a(J, T, JRa) = 9/2 · m4 − 3/2m2; a(J, T, JSa) = 9/2 · m4 + 3/2 · m2, 4.3. THE REE GROUPS 87 where m = 3k. The equality a(J, T −1, z) = a(J, T, z−1) proved in Lemma 1.2.6 then completes this proof.

Lemma 4.3.11. Let x ∈ C (T ) or C (T −1), and C (y) ∈ S. Then Cb(x) · Cb(y) = 0.

2 Proof. From Table 4.3.1, |CG(x)| = 2q and |CG(y)| ≡ ±1 mod 3.

q3(q−1)(q3+1) P χ(x)χ(y)χ(z−1) a(x, y, z) = 2q2·±1 χ(1) χ∈Irr(G) ! P χ(x)χ(y)χ(z−1) P χ(x)χ(y)χ(z−1) ≡ ±q · χ(1) + χ(1) χ∈Irr(G)\{ξ3,ξ4} χ∈{ξ3,ξ4} ! P χ(x)χ(y)χ(z−1) P 0·χ(y)·χ(z−1) = ±q χ(1) + χ(1) ≡ 0 χ∈Irr(G)\{ξ3,ξ4} χ∈{ξ3,ξ4} where the equivalences are take modulo J(O)G. The last equivalence follows from

−1 the fact that ξ3,4(T ) = ξ3,4(T ) = 0 and q does not divide χ(1) for all χ ∈ Irr(G) \

{ξ3, ξ4}.

Using Proposition 1.2.7

Proposition 1.2.7 can be used to compute the multiplications involving the conjugacy

3 class C (X). Note that since the conjugacy class has maximal defect, i.e. |CG(X)| = q and dX = 3(2m + 1), the analysis used in the previous lemmas cannot be applied.

a a a a Lemma 4.3.12. Let y ∈ S = {R ,S ,Vi,Wi,JR ,JS }. Then Cb(X) · Cb(y) = eξ3 − Cb(y). In particular   0 if y = z ∈ V ,W ,Sa,JSa;  i i   −2 if y = z ∈ Ra,JRa; a(X, y, z) =  ±1, if y 6= z and z ∈ S;    0, if z 6∈ S.

Hence (1 + Cb(X)) · Cb(y) = eξ3 .

Proof. Recall that S consists of q − 2 conjugacy classes. For y in S we have so far calculated all structure constants a(x, y, z) except a(X, y, z). Hence we can use 88 CHAPTER 4. SUZUKI AND REE GROUPS

Proposition 1.2.7 to ﬁnd these remaining ones. All equivalences are taken modulo J(O)G. P We have |C (y)| ≡ 0. Hence by Proposition 1.2.7, |C (y)| = x a(x, y, z) ≡ 0. Let α = a(X, y, z). Then

X a(x, y, z) = a(1G, y, z) + a(J, y, z) + (q − 2)a(x ∈ S, y, z) + α x

 a a  1 + 1 + (q − 2)(+1) + α ≡ α, if y = z ∈ Vi,Wi,S ,JS ;   a a  1 − 1 + (q − 2)(−1) + α ≡ 2 + α, if y = z ∈ R ,JR ;  a a = 0 + 1 + (q − 2)(+1) + α ≡ −1 + α if y 6= z and z ∈ Vi,Wi,S ,JS ;   0 + (−1) + (q − 2)(−1) + α ≡ α + 1 if y 6= z and z ∈ Ra,JRa;    α, if z 6∈ S.

Lemma 4.3.13. Let y ∈ C (J). Then Cb(X) · Cb(J) = eξ3 − Cb(J).

P Proof. By Proposition 1.2.7, x a(x, J, z) = |C (J)| = |G|/((q+1)q(q−1)) ≡ 0 mod 3. Hence

X a(x, J, z) = a(1G, J, z) + a(J, J, z) + (q − 2)a(x ∈ S, J, z) + a(X, J, z) ≡ 0 x   1 + 0 + (q − 2)(0) + a(X, J, z) ≡ 1 + a(X, J, z), if z ∈ J;   a a  0 + 1 + (q − 2)(+1) + a(X, J, z) ≡ −1 + a(X, J, z), if z ∈ Vi,Wi,S ,JS ; =  0 − 1 + (q − 2)(−1) + a(X, J, z) ≡ 1 + a(X, J, z), if z ∈ Ra,JRa;    a(X, J, z), if z 6∈ S, C (J).

Lemma 4.3.14. Let x ∈ C (X) and y ∈ C (YT ), C (YT −1), C (JT ) or C (JT −1). Then Cb(x) · Cb(y) = 2 · Cb(y).

Proof. Fix y ∈ C (YT ); the remaining three cases are proved in the same way. P By Proposition 1.2.7, x a(x, Y T, z) = |C (YT )| = |G|/(3q) ≡ 0 mod 3. Hence ! X a(x, Y T, z) + a(X, Y T, z) ≡ 0 mod J(O)G for all z ∈ G. x6=X 4.3. THE REE GROUPS 89

Now a(x 6= X, Y T, z) ≡ 0 except a(1G,YT,YT ) = 1. Hence   0, if z 6∈ C (YT ); a(X, Y T, z) =  2, if z ∈ C (YT ).

Finding the remaining structure constants by “brute force”

For the remaining structure constants, direct calculations from the character table [War66] are used. The exact values can be found in Appendix B; here only the values modulo J(O)G are given. We make use of Lemma 1.2.6 to reduce the number of explicit calculations needed.

Cb(T ) · Cb(YT −1) = 0 = Cb(T −1) · Cb(YT ) Cb(T ) · Cb(YT ) = 0 = Cb(T −1) · Cb(YT −1) Cb(T ) · Cb(JT −1) = 0 = Cb(T −1) · Cb(JT ) Cb(T ) · Cb(JT ) = 0 = Cb(T −1) · Cb(JT −1) Cb(Y ) · Cb(T ) = 0 = Cb(Y ) · Cb(T −1)

Cb(Y ) · Cb(X) = 2 · Cb(Y ) P a P a P P Cb(X) · Cb(X) = 2 + Cb(X) + Cb(R ) + Cb(S ) + Cb(Vi) + Cb(W ) P a P a P a Cb(T ) · Cb(T ) = 2 · a Cb(R ) + a Cb(JR ) + 2 · a Cb(JS ) = Cb(T −1) · Cb(T −1) = Cb(T ) · Cb(T −1) Cb(X) · Cb(T ) = 2Cb(T ) + 2 P Cb(Ra) + P Cb(JRa) + 2 P Cb(JSa) Cb(X) · Cb(T −1) = 2Cb(T −1) + 2 P Cb(Ra) + P Cb(JRa) + 2 P Cb(JSa) 90 CHAPTER 4. SUZUKI AND REE GROUPS

Summary of multiplications

Table 4.3.4: Summary of multiplications of two conjugacy class sums in kG

a a −1 −1 −1 a a R S Vi Wi XYTT YTYT JTJT JR JS J a R eξ3 eξ3 eξ3 eξ3 γ1 − − − − − − − eξ3 eξ3 eξ3 a S eξ3 eξ3 eξ3 γ2 − − − − − − − eξ3 eξ3 eξ3

Vi eξ3 eξ3 γ3 − − − − − − − eξ3 eξ3 eξ3

Wi eξ3 γ4 − − − − − − − eξ3 eξ3 eξ3

X α δ1 µ ν δ2 δ3 δ4 δ5 γ5 γ6 γ7 Y − − − − − − − − − − T β β − − − − − − − T −1 β − − − − − − − YT − − − − − − − YT −1 − − − − − − JT − − − − − JT −1 − − − −

a JR eξ3 eξ3 eξ3 a JS eξ3 eξ3

J eξ3

Here “ − ” denotes a zero in kG and

P a P a P P α = 2 + Cb(X) + Cb(R ) + Cb(S ) + Cb(Vi) + Cb(Wi) β = 2 P Cb(Ra) + P Cb(JRa) + 2 P Cb(JSa)

a γ1 = eξ3 − Cb(R ) δ1 = 2 · Cb(Y ) a γ2 = eξ3 − Cb(S ) δ2 = 2 · Cb(YT ) −1 γ3 = eξ3 − Cb(Vi) δ3 = 2 · Cb(YT )

γ4 = eξ3 − Cb(Wi) δ4 = 2 · Cb(JT ) a −1 γ5 = eξ3 − Cb(JR ) δ5 = 2 · Cb(JT ) a P a P a P a γ6 = eξ3 − Cb(JS ) µ = 2Cb(T ) + 2 Cb(R ) + Cb(JR ) + 2 Cb(JS ) −1 P a P a P a γ7 = eξ3 − Cb(J) ν = 2Cb(T ) + 2 Cb(R ) + Cb(JR ) + 2 Cb(JS )

Note that Cb(X)Cb(y) = γi = eξ3 − Cb(y) so that (1 + Cb(X))Cb(y) = eξ3 . Moreover,

C (X) · C (Y ) = δi = 2 · Cb(y) so that (1 + Cb(X))Cb(y) = 0. 4.3. THE REE GROUPS 91

Since eξ3 · e0 = 0, we can see from Table 4.3.4 that most pairs of elements in DG 0 0 multiply to zero. However there exist elements b, b in DG such that b · b 6= 0:

2 P a P a P P (1 + Cb(X)) e0 = Cb(R ) + Cb(S ) + Cb(Vi) + Cb(Wi) e0

= 2 P Cb(Ra) + P Cb(JRa) + 2 P Cb(JSa)

−1 ±1 2 −1 (1 + Cb(X))Cb(T )e0 = (1 + Cb(X))Cb(T )e0 = (Cb(T )) e0 = Cb(T ) · Cb(T )e0

= 2 P Cb(Ra) + P Cb(JRa) + 2 P Cb(JSa)

Proof of Theorem 4.3.1. By Table 4.3.4 and the discussion below it, there exist

0 0 elements b, b in DG such that b · b 6= 0. Hence LL(Z(kGe0)) ≥ 3. Note that the number of the conjugacy classes labeled by C (Sa), for some a, is the only one not congruent to zero modulo 3; in fact, we have (q − 3)/24 ≡ 1

2 2 modulo 3 of those (see Table 4.3.1). This explains why (1 + Cb(X)) e0 6= (1 + Cb(X)) ±1 ±1 ±1 2 ±1 2 while (1 + Cb(X))Cb(T )e0 = (1 + Cb(X))Cb(T ), and (Cb(T )) e0 = (Cb(T )) . In particular,

(q−3)/24  (q−3)/24  (q−3)/24  X a X a X a  Cb(S ) · eξ3 ≡  Cb(S ) ·  Cb(S ) ≡ eξ3 6= 0. a=1 a=1 a=1

From the multiplications already computed, it can be concluded that the Loewy length must be equal to 3, since none of the outcomes of the non-zero multiplications

−1 a of two elements in DG involve the conjugacy classes C (X), C (T ), C (T ) or C (S ).

It therefore follows that any triple of elements in DG will multiply to zero. Hence

LL(Z(kGe0)) = 3. 92 CHAPTER 4. SUZUKI AND REE GROUPS

4.3.4 The Normaliser of a Sylow 3-subgroup

In this section, our focus will be on the normaliser of a Sylow 3-subgroup P of G. In order to compute the Loewy length of the centre of the group algebra kNG(P ), the conjugacy classes and character table of NG(P ) needs to be calculated ﬁrst; this will then allow the use of Burnside’s formula. A considerable amount of work will go into the computation of the character table, which is given in full in Theorem 4.3.15. Note that a partial character table was computed by Gramain [Gra05], which will form the starting point for the work presented in this section.

The Conjugacy classes of NG(P )

As the conjugacy class sums form a basis for the centre of the group algebra, ﬁrst the conjugacy classes of N = NG(P ), and corresponding centralisers need to be computed. The main reference for the initial construction of the group N is the description given in [Car89] and [Gra05]. ∼ Let P be a Sylow 3-subgroup of G. Then P = {x(t, u, v) | t, u, v ∈ Fq} with multiplication

3θ 3θ+1 2 3θ x(t1, u1, v1)x(t2, u2, v2) = x(t1 + t2, u1 + u2 − t1t2 , v1 + v2 − t2u1 + t1t2 − t1t2 )

θ 3k where θ is the automorphism of Fq given by λ = λ for all λ ∈ Fq. The inverse of an element x(t, u, v) ∈ P is given by

x(t, u, v)−1 = x(−t, −u − t3θ+1, −v − tu + t3θ+2).

There exists a complement W to P in NG(P ), i.e. NG(P ) = PW , |W | = q − 1, × and P ∩ W = 1. The cyclic group W can be denoted by the set {h(w) | w ∈ Fq } and conjugation of P by W is given by

h(w)x(t, u, v)h(w)−1 = x(w2−3θt, w3θ−1u, wv)

where t, u, v, w ∈ Fq, w 6= 0. Note that h(1) = 1W = 1G and h(−1) is the unique involution in W . 4.3. THE REE GROUPS 93

The elements of P form 7 conjugacy classes in NG(P ) [Gra05, Section 2.5.2].

label of element elements order |CN (g)| |C (g)|

1P x(0, 0, 0) 1 |G| 1 3 Z(P ) \{1P } x(0, 0, v), v 6= 0 3 q q − 1 T,T −1 x(0, u, v), u 6= 0 3 2q2 q(q − 1)/2 Y,YT,YT −1 x(t, u, v), t 6= 0 9 3q q2(q − 1)/3

Since there are q + 7 conjugacy classes overall in NG(P ), the remaining elements of the form x(t, u, v)h(w), w 6= 1, must fall into q conjugacy classes. The next task is to distinguish these q conjugacy classes, and to show that they are as follows:

element order |CN (g)| |C (g)| P h(w), w 6= ±1 |h(w)| q − 1 q3 x(0, 0, 0)h(−1) 2 q(q − 1) q2 x(0, −1, 0)h(−1) 6 2q q2(q − 1)/2 (x(0, −1, 0)h(−1))−1 6 2q q2(q − 1)/2

Finding CN (h(w)) for w 6= 1 −1 Let x ∈ CP (h(w)) then h(w) ∈ CN (x). Thus x(t, u, v) = h(w)x(t, u, v)h(w) = x(w2−3θt, w3θ−1u, wv).

As w 6= 1, for x(t, u, v) to be in CP (h(w)),

• v = 0;

2−3θ × • w = 1 and t ∈ Fq ;

3θ−1 × • w = 1 and u ∈ Fq .

 2−3θ  {x(t, 0, 0) | t ∈ Fq}, if w = 1 size q;  3θ−1 CP (h(w)) = {x(0, u, 0) | t ∈ Fq}, if w = 1 size q;   {x(0, 0, 0)}, otherwise.

2−3θ However note that w = 1 has a unique solution w ∈ Fq: w = 1; furthermore 3θ−1 w = 1 has two solutions w ∈ Fq: w = ±1 [Gra05, p. 62]. Hence if w = −1, then CP (h(−1)) = {x(0, u, 0) | u ∈ Fq} which has size q. Moreover as W is abelian,

CN (h(−1)) = CP (h(−1)) · W and so |CN (h(−1))| = q(q − 1). Finally for w 6= ±1,

CN (h(w)) = x(0, 0, 0)W and |CN (h(w))| = q − 1. 94 CHAPTER 4. SUZUKI AND REE GROUPS

Finding the conjugacy classes of elements from W in N

3 Note that for w 6= ±1, as |CN (h(w))| = q − 1, then |C (h(w))| = q . Consider

x(t, u, v)h(w)x(t, u, v)−1 = x(t, u, v)x(t0, u0, v0)h(w),

0 0 0 for some t, t , u, u , v, v ∈ Fq. Then C (h(w)) = A · h(w) for some A ⊆ P ; hence |C (h(w))| = |A|. In particular, if w 6= ±1, then |A| = |P | and thus C (h(w)) = P h(w). Now assume w = −1,

x(t, u, v)h(w)h(−1)h(w)−1x(t, u, v)−1 = x(t, u, v)h(−1)x(t, u, v)−1 = x(t, t3θ+1, v + tu)h(−1)

2 As we can freely choose t ∈ Fq and then (v, u) such that v+tu gives Fq, the q elements x(t, t3θ+1, v + tu)h(−1) form the conjugacy class C (h(−1)). However, it is clear that x(0, −1, 0)h(−1) 6∈ C (h(−1)).

Next take x(0, −1, 0)h(−1) ∈ NG(P ). What is CN (x(0, −1, 0)h(−1))?

As before, g ∈ CN (x(0, −1, 0)h(−1)) implies that x(0, −1, 0)h(−1) ∈ CN (g). Thus

g = x(0, −1, 0)h(−1)x(t, u, v)h(w)h(−1)x(0, −1, 0)−1 = x(−t, −1 + u + w3θ−1, −v − t)h(w)   −t = t, ⇒ t = 0;  3θ−1 3θ−1 = x(t, u, v)h(w) ⇒ −1 + u + w = u, w = 1 ⇒ w = ±1 , u ∈ Fq;   −v − t = v, ⇒ v = 0.

Hence CN (x(0, −1, 0)h(−1)) = {x(0, u, 0)h(w) ∈ NG(P ) | u ∈ Fq, w = ±1} which has size 2q.

Which elements are in the conjugacy class of x(0, −1, 0)h(−1)?

x(t, u, v)h(w)x(0, −1, 0)h(−1)h(w)−1x(t, u, v)−1 = x(t, u, v)x(0, −w3θ−1, 0)h(−1)x(t, u, v)−1 = x(−t, −w3θ−1 + t3θ+1, −v − t3θ+2 + tw3θ−1)h(−1)

Note that (x(0, −1, 0)h(−1))−1 = h(−1)x(0, 1, 0) = x(0, 1, 0)h(−1). By the above calculation, (x(0, −1, 0)h(−1))−1 6∈ C (x(0, −1, 0)h(−1)), otherwise t = 0 and −w3θ+1 = × 1, which has no solution w ∈ Fq . Hence the two remaining conjugacy classes are an inverse pair. 4.3. THE REE GROUPS 95

Summary of conjugacy classes in NG(P )

The elements of NG(P ) form the following q + 7 conjugacy classes. As before, t, u, v ∈ × Fq and w ∈ Fq .

Table 4.3.5: Conjugacy classes of NG(P )

label C (g) o(g) |CN (g)| CN (g) |C (g)| 1N x(0, 0, 0)h(1) 1 |N| NG(P ) 1 3 X = Z(P ) \{1P } x(0, 0, v)h(1) 3 q P q − 1 v 6= 0 T,T −1 x(0, u, v)h(1) 3 2q2 x(0, u, v)h(w) q(q − 1)/2 u 6= 0 w = ±1 −1 2 Y,YT,YT x(t, u, v)h(1) 9 3q x(0, 0, v2), q (q − 1)/3 t 6= 0 x(t, u, v2), 3θ+1 x(−t, −t − u, v2) where v2 ∈ Fq P h(w) x(t, u, v)h(w) |h(w)| q − 1 x(0, 0, 0)h(w) q3 w 6= ±1 w ﬁxed , w 6= ±1 J : x(0, 0, 0)h(−1) x(t, t3θ+1, v + tu)h(−1) 2 q(q − 1) x(0, u, 0)h(w) q2 JT α 6 2q x(0, u, 0)h(w) q2(q − 1)/2 x(0, −1, 0)h(−1) w = ±1 JT −1 β 6 2q x(0, u, 0)h(w) q2(q − 1)/2 (x(0, −1, 0)h(−1))−1 w = ±1

where

α = x(−t, −w3θ−1 + t3θ+1, −v − t3θ+1 + tw3θ−1)h(−1)

and

β = x(−t, w3θ−1 + t3θ+1, −v − t3θ+1 − tw3θ−1)h(−1).

Detailed construction of the character table

Gramain [Gra05, Section 2.5.4] gives the following characters of NG(P ) which are induced from characters in P . Irr(P) Irr(N)

{λ2, . . . , λq} → λ, degree q − 1

{ψ2, . . . , ψq} → ψ, degree (q − 1)q q−1 k {χi,j, i = 1, 2 and 1 ≤ j ≤ 2 } → χ, degree (q − 1)3 q−1 k {χi,j, i = 1, 2 and 1 ≤ j ≤ 2 } → χ, degree (q − 1)3 q−1 q−1 k {χ3,j, 1 ≤ j ≤ 2 } → µ1, µ2, degree 2 3 q−1 q−1 k {χ3,j, 1 ≤ j ≤ 2 } → µ1, µ2, degree 2 3 96 CHAPTER 4. SUZUKI AND REE GROUPS

Moreover NG(P ) has q − 1 linear irreducible characters α0 = 1N , . . . , αq−2 given by

αi = αei ◦ π where π : NG(P ) = PW → W is the natural homomorphism and N Pq−2 Irr(W ) = {αei, 0 ≤ i ≤ q − 2}. We have IndP (1P ) = i=1 αi. Hence the characters αi have P in their kernel, so that αi(g) = 1 for all 0 ≤ i ≤ q − 2 and g ∈ P .

The following table summarises which parts of the character table are known; see Table 4.3.5 for the conjugacy classes, Eaton [Eat00] for the character degrees and Gramain [Gra05] for the character values on elements of order 9.

2 2 2 2 q(q−1) q (q−1) q (q−1) q (q−1) 3 2 q (q−1) |C (g)| 1 q − 1 2 3 3 3 q q 2

3 3 2 |CN (g)| q (q − 1) q 2q 3q 3q 3q q − 1 q(q − 1) 2q

1 XT,T −1 YYTYT −1 P h(w) JJT,JT −1

α0 = 1N 1 1 1 1 1 1 1 1 1

α1, . . . , αq−2 1 1 1 1 1 1 λ q − 1 −1 −1 −1 k 3 (q−1) k k k µ1 2 −ε3 −ε3 ω −ε3 ω k 3 (q−1) k k k µ2 2 −ε3 −ε3 ω −ε3 ω k 3 (q−1) k k k µ1 2 −ε3 −ε3 ω −ε3 ω k 3 (q−1) k k k µ2 2 −ε3 −ε3 ω −ε3 ω χ 3k(q − 1) ε3k ε3kω ε3kω χ 3k(q − 1) ε3k ε3kω ε3kω ψ q(q − 1) 0 0 0

where q = 32k+1, ε ∈ {±1} and ω = e2iπ/3.

The aim is to ﬁll in the remaining entries; for this we apply the orthogonality relations of a character table, given in Theorem 1.2.2.

Pq The character values of ψ We have ψ|P = i=2 ψi and ψi(g) = 0 for all g ∈ P \ Z(P ) [Gra05, p.70]. Hence ψ(g) = 0 for all g ∈ P \Z(P ). Furthermore since ψ is an induced N character, ψ = IndP ψi, ψ(g) = 0 for all g ∈ N \ P . This leaves ψ(X) as the only unknown value. 4.3. THE REE GROUPS 97

Apply the orthogonality relation to ψ and α0.

1 P hψ, α0i = |N| g∈N ψ(g)α0(g)

1 = |N| (q(q − 1) + ψ(X)(q − 1)) = 0

⇒ ψ(X) = −q

N The character values of λ Since λ is an induced character, λ = IndP λi, λ(g) = 0 for all g ∈ N \ P . To determine λ(X) apply the orthogonality relation to λ and ψ.

1 P hλ, ψi = |N| g∈N λ(g)ψ(g)

1 = |N| (q(q − 1)(q − 1) − qλ(X)(q − 1)) = 0

⇒ λ(X) = q − 1

Let λ(T ) = a, hence λ(T −1) = λ(T ) = a. In fact since k ≥ 1, λ is the only character of degree q − 1 and so it is real-valued, i.e. a = a. Now apply an orthogonality relation.

1 P hλ, α0i = |N| g∈N λ(g)α0(g)

1 2 q(q−1) q(q−1) q2(q−1) q2(q−1) q2(q−1) = |N| q − 1 + (q − 1) + a 2 + a 2 − 3 − 3 − 3 = 0

⇒ 2a = 2(q − 1) ⇒ λ(T ) = λ(T −1) = q − 1

N The character values of χ Since χ is an induced character, χ = IndP χj, χ(g) = 0 for all g ∈ N \ P . To determine χ(X) apply the orthogonality relation to χ and ψ.

1 P hχ, ψi = |N| g∈N χ(g)ψ(g)

1 k = |N| q(q − 1)3 (q − 1) − q(q − 1)χ(X) = 0

⇒ χ(X) = 3k(q − 1)

Let χ(T ) = b, hence χ(T −1) = χ(T ) = b. Now apply orthogonality relations.

1 P hχ, α0i = |N| g∈N χ(g)α0(g)

1 k k 2 q(q−1) q(q−1) q2(q−1) k = |N| 3 (q − 1) + 3 (q − 1) + b 2 + b 2 + 3 · ε · 3

q2(q−1) k q2(q−1) k + 3 · ε · 3 · ω + 3 · ε · 3 · ω

1 k q(q−1) = |N| 3 q(q − 1) + 2 (b + b) = 0

⇒ b + b = −2 · 3k 98 CHAPTER 4. SUZUKI AND REE GROUPS

1 P hχ, χi = |N| g∈N χ(g)χ(g)

1 k 2 k 2 q(q−1) q(q−1) = |N| (3 (q − 1)) + (3 (q − 1)) (q − 1) + bb 2 + bb 2

q2(q−1) k 2 q2(q−1) k 2 q2(q−1) k 2 + 3 · (ε · 3 ) + 3 · (ε · 3 ) · ωω + 3 · (ε · 3 ) · ωω

→ bb = 32k(32k+1 + 1)

Now since (x + b)(x + b) = x2 + (b + b)x + bb = x2 + (−2 · 3k)x + 32k(32k+1 + 1) we can use the quadratic formula to determine b and b.

√ √ k −b± b2−4ac 2·3 ± 4·32k−4(32k(32k+1+1)) 2a = 2 √ = 3k ± 32k −3 √ √ Hence χ(T ) = −3k + 32k −3 and χ(T −1) = −3k − 32k −3. Finally note that χ(g) = χ(g) so we have determined two more characters.

So far we have calculated the following additional entries:

2 2 q(q−1) q(q−1) 3 2 q (q−1) q (q−1) |C (g)| q − 1 2 2 q q 2 2

3 2 2 |CN (g)| q 2q 2q q − 1 q(q − 1) 2q 2q

XTT −1 P h(w) JJTJT −1

α0 = 1N 1 1 1 1 1 1 1

α1, . . . , αq−2 1 1 1 λ q − 1 q − 1 q − 1 0 0 0 0 √ √ χ 3k(q − 1) −3k + 32k −3 −3k − 32k −3 0 0 0 0 √ √ χ 3k(q − 1) −3k − 32k −3 −3k + 32k −3 0 0 0 0 ψ −q 0 0 0 0 0 0 where q = 32k+1, ε ∈ {±1} and ω = e2iπ/3.

The characters αi We need to determine αi(g) for g ∈ N \ P .

Recall that α0 = 1N , . . . , αq−2 are the q − 1 linear irreducible characters of N given by αi = αei ◦ π where π : NG(P ) = PW → W is the natural homomorphism and ∼ Irr(W ) = {αei, 0 ≤ i ≤ q − 2}. The character table of W = Cq−1 consists of powers of the primitive (q − 1)th root of unity. 4.3. THE REE GROUPS 99

∼ Table 4.3.6: Character table of W = hhi = Cq−1

1 h h2 h3 . . . h(q−1)/2 . . . hq−3 hq−2 αe0 1 1 1 1 ... 1 ... 1 1 2 3 q−3 q−2 αe1 1 ξ ξ ξ −1 ξ ξ 2 4 6 2(q−3) 2(q−2) αe2 1 ξ ξ ξ 1 ξ ξ . . q−2 2(q−2) 3(q−2) (q−2)(q−3) (q−2)(q−2) αeq−2 1 ξ ξ ξ −1 ξ ξ where ξ is a primitive (q − 1)th root of unity.

(q−1)/2 The unique involution h ∈ W corresponds to J ∈ NG(P ). We now have  i (q−1)/2  1, if i is even, or i = 0; αi(J) = αei(J) = (ξ ) =  −1, if i is odd.

±1 ±1 αi(JT ) = αei(JT · P ) = αei(J · P ) = αei(J) and

j ij αi(P h(w)) = αei(h ) = ξ .

(q−1)/2 −1 Note that P h splits into the three conjugacy classes J, JT and JT in NG(P ).

In summary, one section of the character table for NG(P ) is given by:

(q−1) (q−1) (q−1) (q−1) W h h2 h3 . . . hj 6= h 2 . . . hq−3 hq−2 h 2 h 2 h 2 l l l l l l l l l l −1 PW P h(w1) P h(w2) P h(w3) . . . P h(wj ) . . . P h(wq−3) P h(wq−2) JJTJT 2 3 j q−3 q−2 α1 ξ ξ ξ ξ ξ ξ −1 −1 −1 2 4 6 2j 2(q−3) 2(q−2) α2 ξ ξ ξ ξ ξ ξ 1 1 1 . . q−2 2(q−2) 3(q−2) (q−2)j (q−2)(q−3) (q−2)(q−2) αq−2 ξ ξ ξ ξ ξ ξ −1 −1 −1 where ξ is a primitive (q − 1)th root of unity.

The character values of µi and µi To determine µi(X) apply the orthogonality relation to µi and ψ.

1 P hµi, ψi = |N| g∈N µi(g)ψ(g)

1 3k(q−1) = |N| q(q − 1) 2 − q(q − 1)µi(X) = 0

3k(q−1) ⇒ µi(X) = 2

3k(q−1) Hence µ1(X) = µ2(X) = µ1(X) = µ2(X) = 2 . 100 CHAPTER 4. SUZUKI AND REE GROUPS

µi(P h(w) = 0), w 6= ±1: Suppose µ1(P h(w)) 6= 0. Then by Lemma 1.2.3, αiµ1 must be an irreducible character of N for all 1 ≤ i ≤ q − 2. However since there are 3k(q−1) only 4 irreducible characters of degree 2 , no elements h(w) of order 4, and q − 1 distinct linear irreducible character (in particular αi(P h(w)) is distinct for each i) this leads to a contradiction.

Hence µ1(P h(w)) = µ2(P h(w)) = µ1(P h(w)) = µ2(P h(w)) = 0 for all conjugacy classes P h(w) where h(w) is not the identity or involution in W , i.e. w 6= ±1.

N µi(J): Let µ1(J) = c. Since J ∈ N \ P , we know that µ1(J) + µ2(J) = IndP θ(J) = −1 0. Furthermore C (J ) = C (J). Hence c ∈ R; and µ2(J) = −c, µ1(J) = c and

µ2(J) = −c. We can now apply the second orthogonality relation to the conjugacy class J.

P χ(J)2 = q − 1 + c2 + (−c)2 + c2 + (−c)2 χ∈Irr(N) = q − 1 + 4c2 = q(q − 1) ⇒ 4c2 = q(q − 1) − (q − 1) ⇒ c2 = (q − 1)2/4 ⇒ c = (q − 1)/2

q−1 q−1 q−1 q−1 Hence µ1(J) = 2 , µ2(J) = − 2 , µ1(J) = 2 and µ2(J) = − 2 .

−1 −1 µi(JT ), µi(JT ): Again, since JT,JT ∈ N\P , we know that µ1(JT )+µ2(JT ) = N IndP θ(JT ) = 0. We have the following part of the character table:

JJTJT −1

a0 = 1N 1 1 1

α1, . . . , αq−2 ±1 ±1 ±1 λ 0 0 0 q−1 µ1 2 m m q−1 µ2 − 2 −m −m q−1 µ1 2 m m q−1 µ2 − 2 −m −m χ, χ, ψ 0 0 0 4.3. THE REE GROUPS 101

We can apply the second orthogonality relation to the conjugacy class JT .

P χ(JT )χ(JT ) = q − 1 + mm + (−m)(−m) + mm + (−m)(−m) χ∈Irr(N) = q − 1 + 4(mm) = 2q q+1 ⇒ mm = 4 Next apply the second orthogonality relation to the conjugacy classes JT and J.

P q−1 q−1 q−1 q−1 χ(JT )χ(J) = q − 1 + m 2 + (−m)(− 2 ) + m 2 + (−m)(− 2 ) χ∈Irr(N) = q − 1 + m(q − 1) + m(q − 1) = 0 ⇒ m + m = −1

Since (x + m)(x + m) = x2 + (m + m)x + mm = x2 + (−1)x + (q + 1)/4, we can use the quadratic formula to determine m and m. √ √ q+1 2 −b± b −4ac 1± 1−4 4 2a = 2 √ 1±3k −3 = 2 √ −1−3k −3 Hence m = 2 .

−1 −1 µ1(T ) and µ1(T ): Let µ1(T ) = s, then µ1(T ) = s; now apply the orthogonality relation to ﬁnd ss and s + s.

1 P hµ1, µ1i = |N| g∈N µ1(g)µ1(g)

1 8k 6k 4k = q3(q−1) 297/4 · 3 − 27 · 3 + 3/4 · 3 + ss(q(q − 1)) = 1

1 4k+1 2k ⇒ ss = 4 (3 + 3 )

1 P hµ1, α0i = |N| g∈N µ1(g) · 1

1 5k 3k q(q−1) = q3(q−1) 9/2 · 3 − 3/2 · 3 + (s + s) 2 = 0

⇒ s + s = −3k

2 2 k 1 4k+1 2k Now since (x + s)(x + s) = x + (s + s)x + ss = x + (−3 )x + 4 (3 + 3 ) we can use the quadratic formula to determine s and s. √ √ k 2k 1 4k+1 2k −b± b2−4ac 3 ± 3 −4· 4 (3 +3 ) 2a = 2 √ 3k±32k −3 = 2 102 CHAPTER 4. SUZUKI AND REE GROUPS

P P P Observe that g∈J µi(g) + g∈JT µi(g) + g∈JT −1 µi(g) = 0 for i ∈ {1, 2}. As −1 −1 hµi, α0i = 0, it follows that µ1(T ) + µ1(T ) = µ2(T ) + µ2(T ). Moreover, µ2(g) = −1 −1 −1 −µ1(g) for g ∈ {J, JT, JT } and µ2(g) = µ1(g) for g ∈ G \{T,T , J, JT, JT }.

Therefore as hµi, µii = 1, for i ∈ {1, 2}, it follows that (up to complex conjugate) −1 −1 µ1(T ) = µ2(T ) and µ1(T ) = µ2(T ). The values on the conjugacy classes T and T −1 are as follows: √ √ −3k + 32k −3 −3k − 32k −3 µ (T ) = µ (T ) = , µ (T −1) = µ (T −1) = 1 2 2 1 2 2 √ √ −3k − 32k −3 −3k + 32k −3 µ (T ) = µ (T ) = , µ (T −1) = µ (T −1) = 1 2 2 1 2 2

Combining all the character values together, we have proven the following theorem.

2 2k+1 Theorem 4.3.15. Let G = G2(q) where q = 3 , and let NG(P ) be the normaliser of a Sylow 3-subgroup. Then the character table of NG(P ) is given by Table 4.3.7. 4.3. THE REE GROUPS 103 3 3 3 3 − − − − 1 √ √ √ √ − q k k k k 2 2 2 2 3 3 2 − − 1+3 1 − +1 − +1+3 3 3 3 3 − − − − √ √ √ √ q k k k k 2 2 2 2 3 3 1 1) − − − 1 1 1 1+3 2 − q − − +1+3 +1 ( − − h 1) 2 1 1 1) 1 1 2 2 − − 1 1 2 − − 2 2 q q − − JJTJT q q q ( − − q − − ( rectangle given by h q q 1) ) × 1 1 2 j − 1 1 1 1 q JJTJT ( − − w 2) 0 0 0 0 00 0 0 0 0 0 0 ( − h − P h ) q 2) 2 − − 2) 2 q 2 q 1 ω ω ω ω − − − ω ω k k k k q − w q 1 0 0 0 0 q 2)( q q k k 3 3 3 3 ( 2( ξ h − 3 3 3 ε ε ε ε is an ( − q ξ ( − − − − ) P h ξ P ( ) ω ω ω ω 3) 3 G ωω ε ε k k k k 1 − q k k − 3) 3 3 3 3 3 q 3 N q 3 3 3 ε ε ε ε − − − − ε ε q w q q 2)( − − − − ( 2( ξ − q ξ ( k k k k ξ k k 3 3 3 3 1 q 3 3 ε ε ε ε YYTYT 1)th root of unity. 3 − ε ε − − − − − . . . h . . . P h q 3 3 1) − − 3 3 3 3 2 ) − j − − − − q j √ √ ( 2) √ √ √ √ k j k 1 w j 1 h k k k k 2 2 2 2 − ( 2 2 2 2 ξ − 2 2 2 2 q q ξ 3 3 3 − ( 2 6 = − − +3 +3 ξ q + 3 − k k k k j 3 3 3 3 k k − − − − 3 3 − − . Moreover the section denoted is a primitive ( 3 . . . h . . . P h ξ 3 3 iπ/ ) 2 − − 3 3 3 3 3 Table 4.3.7: Character table of e 2) − − − − √ √ w − 3 3 6 √ √ √ √ k k 1 q ( = ξ ξ k k k k h 2 2 2 where 2 2 2 2 3( 2 2 2 2 q 3 0 0 0 0 0 0 0 0 0 3 3 − ω ξ 2 P h +3 +3 − − q + 3 − k k k k 3 3 3 3 k k ) − − − − 3 and 3 2 2) w − − − 2 } 2 4 q ( ξ 1 2( ξ P h 1) 1) 1) 1) 1) 1) 1 q − − − − − − 3 ∈ {± ) 2 2 2 2 q q q q − q XTT ( ( ( ( 1 − q q ε 2 k k k k ( ( q w 3 3 3 3 2 , − k k ξ ξ l l l l l l l l l ( h h q ξ ξ +1 k P h 2 1) 3 1) 3 1) 1) 1) 1) 1) 1) 1 2 = 3 − − − − − − − 1 2 − 2 2 2 2 − 1 111 1 1 1 11111111 1 1 1 1 q q q q . . . − ( ( ( ( q q q q q α α q k k k k ( ( ( q α ( 3 3 3 3 3 k k q q 3 3 2 where − | q N ) g 2 1 2 1 ( λ χ χ ψ = 1 µ µ µ µ N 0 C , . . . , α | α 1 α 104 CHAPTER 4. SUZUKI AND REE GROUPS

Main theorem concerning the normaliser in small Ree groups

We now state the main theorem on the normaliser of a Sylow 3-subgroup. Note that the group algebra kN is indecomposable, where k is an algebraically closed ﬁeld of characteristic 3, so we don’t need to consider any block idempotents.

2 2k+1 Theorem 4.3.16. Let N = NG(P ) where G = G2(q), q = 3 ≥ 27, and P ∈

Syl3(G). Then LL(Z(kN)) = 2.

By Table 4.3.5, all non-trivial conjugacy classes of N have class size divisible by 3 except C (X) which has size |C (X)| = q − 1. Hence by Proposition 1.5.10, a basis for J(Z(kN)) is given by

BN = {Cb(x) | x ∈ P, x 6= 1N , x 6∈ C (X)} ∪ {Cb(X) + 1}.

Similar to the work in G presented in Section 4.3.3, the proof of Theorem 4.3.16 will spread over several lemmas. Recall that cc(N) denotes the set of conjugacy classes inside N. All conjugacy class sums are multiplied as elements in kN, and all equivalences are taken modulo J(O)N.

Lemma 4.3.17. Let C (x) ∈ {P h(wj)} and C (y) ∈ cc(N) \{Cb(1N ), Cb(X)}. Then Cb(x) · Cb(y) = 0.

3 Proof. By Table 4.3.5, |CG(x)| = q − 1 and |CG(y)|3 < q . Hence for all z ∈ N,

3 −1 a(x, y, z) = q (q−1) P χ(x)χ(y)χ(z ) (q−1)·|CG(y)| χ(1) χ∈Irr(G)

q−2 a 1 P −1 ≡ 3 · s αi(x)αi(y)αi(z ) where a ≥ 1, gcd(3, s) = 1 i=1

≡ 0

Lemma 4.3.18. Let x ∈ C (J), C (JT ) or C (JT −1), and C (y) ∈ cc(N) \

{C (1N ), C (X), C (P h(wj))}. Then Cb(x) · Cb(y) = 0.

Proof. Suppose y ∈ C (J), C (JT ) or C (JT −1).

q3(q−1) P χ(x)χ(y)χ(z−1) a(x, y, z) = q2·s χ(1) where gcd(3, s) = 1 χ∈Irr(G) ! 2k+1 1 P P 2·χ(x)χ(y)χ(z−1) = 3 · s ±1 + 3k(q−1) ≡ 0 mod J(O)N αi χ∈{µi,µi} 4.3. THE REE GROUPS 105

Suppose y ∈ C (Y ), C (YT ) or C (YT −1).

q3(q−1) P χ(x)χ(y)χ(z−1) a(x, y, z) = 3q2·s χ(1) where gcd(3, s) = 1 χ∈Irr(G) ! 2k 1 P P 2·χ(x)χ(y)χ(z−1) = 3 · s ±1 + 3k(q−1) ≡ 0 αi χ∈{µi,µi}

Here the last equivalence follows since k ≥ 1.

Suppose y ∈ C (T ) or C (T −1). By applying Theorem 1.2.5, we have

a(x, y, z) ≡ 0 for C (z) ∈ {C (X), C (T ), C (T −1), C (Y ), C (YT ), C (YT −1)}.

For C (z) ∈ {C (J), C (JT ), C (JT −1)} we ﬁnd a(x, y, z) by direct calculation.

a(J, T, z) a(J, T, J) 0 a(J, T, JT ) 0 a(J, T, JT −1) 3m2

a(JT, T, z) a(JT −1, T, z) a(JT,T,J) 9/2m4 − 3/2m2 a(JT −1,T,J) 0 a(JT,T,JT ) 9/4m4 − 9/4m2 a(JT −1,T,JT ) 9/4m4 + 3/4m2 a(JT,T,JT −1) 9/4m4 − 9/4m2 a(JT −1,T,JT −1) 9/4m4 − 9/4m2

Here m = 3k (q = 3m2) and so all structure constants given in the tables are zero modulo 3.

It remains to consider z ∈ C (P h(wj)). By looking at the character table, Table

4.3.7, we see that only the linear characters αi need to be considered. Now αi(T ) = 1, −1 i a αi(J) = αi(JT ) = αi(JT ) = (−1) , and αi(P h(w)) is of the form ξ for ξ a primitive (q − 1)th root of unity and 0 ≤ a ≤ q − 2. Hence the summation in Burnside’s formula reduces to 1 − ξj + ξ2j − ξ3j + ... − ξ(q−2)j. By column orthogonality of Table 4.3.6, this sum is equal to zero and hence a(x, y, z) = 0 in this case. Finally, by Lemma 1.2.6, a(x, y, z) = a(x−1, y−1, z−1), hence a(J, T −1, z−1) = a(J, T, z) and so on; therefore the result automatically holds if y ∈ C (T −1).

Lemma 4.3.19. Let C (x), C (y) ∈ {C (Y ), C (YT ), C (YT −1)}. Then Cb(x)·Cb(y) = 0. 106 CHAPTER 4. SUZUKI AND REE GROUPS

Proof. By Table 4.3.5, |CG(x)| = |CG(y)| = 3q.

q3(q−1) P χ(x)χ(y)χ(z−1) a(x, y, z) = 32q2 χ(1) χ∈Irr(G)

2k−1 P χ(x)χ(y)χ(z−1) = 3 (q − 1) χ(1) where k ≥ 1 χ∈Irr(G)

≡ 0

k Note that for χ1 ∈ {µ1, µ2, µ1, µ2, χ, χ}, we have 3 | χ1(1), however at the same time k k 3 | χ1(x) and 3 | χ1(y). Furthermore while ψ(1) = q(q − 1), the character ψ is zero on x and y, i.e. ψ(x) = ψ(y) = 0. Hence a(x, y, z) ≡ 0 for all z ∈ N.

Lemma 4.3.20. Let C (x) ∈ {C (T ), C (T −1)} and C (y) ∈ {C (Y ), C (YT ), C (YT −1)}. Then Cb(x) · Cb(y) = 0.

2 Proof. By Table 4.3.5, |CG(x)| = 2q and |CG(y)| = 3q.

q3(q−1) P χ(x)χ(y)χ(z−1) a(x, y, z) = 2·3·q3 χ(1) χ∈Irr(G)

P χ(x)χ(y)χ(z−1) If z ∈ C (P h(wj)) then χ(1) = 0 as the only contributions come from χ∈Irr(G) the linear characters, and we are summing over all the (q − 1)th roots of unity.

If z ∈ C (J), C (JT ) or C (JT −1) then the contribution from the linear characters is Pq−2 −1 q−1 q−1 zero: i=0 αi(x)αi(y)αi(z ) = 2 · 1 + 2 · (−1) = 0. Moreover µ1(x) = µ2(x),

µ1(y) = µ2(y) and µ1(z) = −µ2(z) (same for µ1, µ2). Hence in this case a(x, y, z) = q3(q−1) 2·3·q3 · 0 = 0.

−1 If z ∈ C (1N ), C (X), C (T ) or C (T ), then by Theorem 1.2.5, a(x, y, z) ≡ 0 mod J(O)N.

If z ∈ C (Y ), C (YT ) or C (YT −1), then ! (q−1) P χ(x)χ(y)χ(z−1) a(x, y, z) = 2·3 (q − 1) + 1 + χ(1) χ∈A ! (q−1) P χ(x)χ(y)χ(z−1) = 2·3 (q − 1) + 1 + 3k(q−1) χ∈A

−1 where A = {µ1, µ2, µ1, µ2, χ, χ}. The multiplication χ(x)χ(y)χ(z ) is of the form 33k · s, where gcd(3, s) = 1. Hence a(x, y, z) = 32k−1 · s ≡ 0. 4.3. THE REE GROUPS 107

Lemma 4.3.21. Let C (x), C (y) ∈ {C (T ), C (T −1)}. Then Cb(x) · Cb(y) = 0.

2 Proof. By Table 4.3.5, |CG(x)| = |CG(y)| = 2q . If z ∈ C (P h(wj)), C (J), C (JT ) or C (JT −1) then a(x, y, z) = 0. This is obvious since the conjugacy classes consist only of elements in P , which is a normal subgroup of N. Hence multiplying these two class sums cannot generate any elements not in P .

If z in C (1N ), C (X) then by Theorem 1.2.5, a(x, y, z) ≡ 0. If C (z) ∈ {C (T ), C (T −1), C (Y ), C (YT ), C (YT −1)} then by Proposition 4.1.3, aN (x, y, z) ≡ aG(x, y, z) ≡ 0 mod J(O)N.

Proposition 4.3.22. Let C (y) in cc(N)\{C (X), C (1N )}. Then (Cb(X)+1)·Cb(y) = 0 and (Cb(X) + 1)2 = 0.

Proof. So far we have already calculated all structure constants apart from a(X, y, z); hence we can use Proposition 1.2.7 to ﬁnd the remaining ones. By Proposition 1.2.7 and the conjugacy class sizes given in Table 4.3.5, we have P x a(x, y, z) = |C (y)| ≡ 0 mod 3. Hence

X a(1, y, z) + a(x, y, z) + a(X, y, z) ≡ 0 ∀z ∈ N.

x6=1N ,X

Now a(x 6= X, y, z) ≡ 0 and a(1, y, z) ≡ 0 except a(1, y, y) = 1. Therefore we get   0, if C (y) 6= C (z); a(X, y, z) =  2, if C (y) = C (z).

Hence Cb(X) · Cb(y) = 2 · Cb(y) and so (Cb(X) + 1) · Cb(y) ≡ 0. P 2 We have 1+Cb(X) = γ∈Z(P ) γ, and therefore (1+Cb(X)) = q(1+Cb(X)) ≡ 0.

We summarise the results obtained in this chapter.

Theorem 4.3.23. Let k be an algebraically closed ﬁeld of characteristic 3 and G =

2 2k+1 G2(q), q = 3 ≥ 27, P ∈ Syl3(G). Then LL(Z(B0(kG))) = 3 > 2 = LL(Z(kNG(P ))); hence ∼ Z(B0(kG)) 6= Z(kNG(P )).

Therefore, by Theorem 2.3.2, there is no perfect isometry, and hence no derived equivalence between B0(kG) and kNG(P ). Chapter 5

More sporadic simple groups

In this chapter we compute the Loewy length and dimension of the radical squared of the centre of the principal block for some sporadic simple groups and their Sy- low p-normalisers; in particular we consider those groups with non-abelian Sylow p-subgroups. Groups with abelian Sylow subgroups are not interesting to us since Brou´e’sconjecture already tells us of a close connection. To classify the sporadic simple groups with non-abelian Sylow p-subgroups, one can use the list of maximal subgroups contained in [CCN+85] to descent to a smaller subgroup in which the Sylow structure is known. However, to make these computations easier, we use the following result contained in [Bro15] or [NST15], which yields a simpler method to read oﬀ this property from the character table of the group.

Lemma. 5.0.1 Let G be a sporadic simple group. Then G has abelian Sylow p- subgroups if and only if |gG| is not divisible by p for all p-elements g ∈ G. The only exceptions are Ru, J4 with p = 3 and T h with p = 5 in which case the Sylow p-subgroups are non-abelian.

If B is a block, we denote by δ(B) a defect group of B; B0 denotes the principal block of G and b0 denotes the principal block of NG(P ) where P ∈ Sylp(G). We will continue to use the notation as in [CCN+85]; in particular m denotes a cyclic group of order m and pn denotes an elementary abelian p-group of order pn. The calculations are carried out using the programs written in GAP, which were used for the computations in Chapter 3 and can be found in Appendix A. If a group has too many conjugacy classes, it takes a very long time to do these calculations. Hence

108 5.1. MATHIEU GROUPS 109 we will restrict to the sporadic simple groups with less than 40 conjugacy classes (we include J4 since this group was already discussed in Section 3.3).

Table 5.0.1: Some small sporadic simple groups [CCN+85]

number of Group Order conj. classes 4 2 M11 2 · 3 · 5 · 11 10 6 3 M12 2 · 3 · 5 · 11 15 7 2 M22 2 · 3 · 5 · 7 · 11 12 7 2 M23 2 · 3 · 5 · 7 · 11 · 23 17 10 3 M24 2 · 3 · 5 · 7 · 11 · 23 26 3 J1 2 · 3 · 5 · 7 · 11 · 19 15 7 3 2 J2 = HJ 2 · 3 · 5 · 7 21 7 5 J3 = HJM 2 · 3 · 5 · 17 · 19 21 21 3 3 J4 2 · 3 · 5 · 7 · 11 · 23 · 29 · 31 · 37 · 43 62 HS 29 · 32 · 53 · 7 · 11 24 McL 27 · 36 · 53 · 7 · 11 24 He = HHM 210 · 33 · 52 · 73 · 17 33 Ru 214 · 33 · 53 · 7 · 13 · 29 36 O0N 29 · 34 · 5 · 73 · 11 · 19 · 31 30

5.1 Mathieu groups

5.1.1 M11, p = 2

Block information:

Block B Defect δ(B) k(B) LL(Z(B)) dim(J 2(Z(B)))

B0 4 SD16 8 5 4

b0 4 SD16 7 4 2

5.1.2 M12, p = 2

Block information:

Block B Defect δ(B) k(B) LL(Z(B)) dim(J 2(Z(B)))

B0 6 D 11 4 3

b0 6 D 16 3 1

∼ 2 ∼ where D = Syl2(4 : D12) = (C4 × C4) o (C2 × C2). 110 CHAPTER 5. MORE SPORADIC SIMPLE GROUPS

5.1.3 M12, p = 3

Block information:

Block B Defect δ(B) k(B) LL(Z(B)) dim(J 2(Z(B)))

B0 3 (C3 × C3) o C3 11 3 2

b0 3 (C3 × C3) o C3 11 3 1

5.1.4 M22, p = 2

Block information:

Block B Defect δ(B) k(B) LL(Z(B)) dim(J 2(Z(B)))

B0 7 D 12 3 2

b0 7 D 17 2 0

∼ 4 ∼ where D = Syl2(2 : A6) = (((C4 × C4): C2): C2): C2.

5.1.5 M23, p = 2

Block information:

Block B Defect δ(B) k(B) LL(Z(B)) dim(J 2(Z(B)))

B0 7 D 15 3 2

b0 7 D 17 2 0

∼ 4 where D = Syl2(2 : A6), i.e. the defect group is isomorphic to the defect group of the principal 2-block of M22.

5.1.6 M24, p = 2

Block information:

Block B Defect δ(B) k(B) LL(Z(B)) dim(J 2(Z(B)))

B0 10 D 26 LL > 4 7

b0 10 D 61 3 6

∼ 6 where D = 2 : (2 × D8). 5.2. JANKO GROUPS 111

5.1.7 M24, p = 3

Block information:

Block B Defect δ(B) k(B) LL(Z(B)) dim(J 2(Z(B)))

B0 3 D 13 3 3

b0 3 D 13 3 2

∼ ∼ 1+2 ∼ where D = Syl3(M12) = 3+ = (C3 × C3) o C3.

5.2 Janko groups

5.2.1 J2, p = 2

Block information:

Block B Defect δ(B) k(B) LL(Z(B)) dim(J 2(Z(B)))

B0 7 D 17 LL > 3 4

b0 7 D 19 3 6

∼ 2 where D = (Q8 ∗ D8) : 2 .

5.2.2 J2, p = 3

Block information:

Block B Defect δ(B) k(B) LL(Z(B)) dim(J 2(Z(B)))

1+2 B0 3 3+ 13 3 4 1+2 b0 3 3+ 13 3 3

5.2.3 J3, p = 2

Block information:

Block B Defect δ(B) k(B) LL(Z(B)) dim(J 2(Z(B)))

B0 7 D 17 LL > 3 4

b0 7 D 19 3 6

∼ 2 where D = (Q8 ∗ D8) : 2 . 112 CHAPTER 5. MORE SPORADIC SIMPLE GROUPS

5.2.4 J3, p = 3

Block information: Block B Defect δ(B) k(B) LL(Z(B)) dim(J 2(Z(B)))

B0 5 D 16 2 0

b0 5 D 16 2 0

∼ 2 1+2 where D = Syl3(3 .(3 ) : 8).

Remark. Since both centres have Loewy length 2, and the same dimension, there exists an isomorphism between Z(B0) and Z(b0).

1 5.2.5 J4, p = 11

Block information: Block B Defect δ(B) k(B) LL(Z(B)) dim(J 2(Z(B)))

1+2 B0 3 11+ 49 3 5 1+2 b0 3 11+ 49 3 4

5.3 The Higman-Sims group

5.3.1 HS, p = 2

Block information: Block B Defect δ(B) k(B) LL(Z(B)) dim(J 2(Z(B)))

B0 9 D 20 LL > 3 6

b0 9 D 35 3 4

∼ 3 where D = Syl2(4 : L3(2)).

5.3.2 HS, p = 5

Block information: Block B Defect δ(B) k(B) LL(Z(B)) dim(J 2(Z(B)))

1+2 B0 3 5+ 17 3 3 1+2 b0 3 5+ 17 3 2

1 The group J4 is too large to do the calculations for the primes p = 2 and p = 3, hence we simply recall the results obtained in Section 3.3. 5.4. THE MCLAUGHLIN GROUP 113 5.4 The McLaughlin group

5.4.1 McL, p = 2

Block information: Block B Defect δ(B) k(B) LL(Z(B)) dim(J 2(Z(B)))

B0 7 D 18 LL > 5 11

b0 7 D 17 2 0 ∼ where D = Syl2(M22).

5.4.2 McL, p = 3

Block information: Block B Defect δ(B) k(B) LL(Z(B)) dim(J 2(Z(B)))

B0 6 D 21 4 5

b0 6 D 21 3 4 ∼ 4 2 where D = Syl3(3 : (3 : Q8)).

5.4.3 McL, p = 5

Block information: Block B Defect δ(B) k(B) LL(Z(B)) dim(J 2(Z(B)))

B0 3 D 19 3 4

b0 3 D 19 3 3 ∼ ∼ 1+2 where D = Syl5(U3(5)) = 5+ . For details see Section 3.2.

5.5 The Held group

5.5.1 He, p = 2

Block information: Block B Defect δ(B) k(B) LL(Z(B)) dim(J 2(Z(B)))

B0 10 D 26 LL > 4 7

b0 10 D 61 3 6 114 CHAPTER 5. MORE SPORADIC SIMPLE GROUPS

∼ 6 where D = 2 : (2 × D8).

Remark. Note the similarity with the Mathieu group M24 in characteristic 2 . This ∼ is not surprising since the two normalisers are in fact isomorphic, i.e. NM24 (P1) = ∼ 6 6 NHe(P2) = 2 : (2 × D8) < 2 : 3.S6 where P1 ∈ Syl2(M24) and P2 ∈ Syl2(He).

5.5.2 He, p = 3

Block information:

Block B Defect δ(B) k(B) LL(Z(B)) dim(J 2(Z(B)))

1+2 B0 3 3+ 13 3 3 1+2 b0 3 3+ 13 3 2

5.5.3 He, p = 7

Block information:

Block B Defect δ(B) k(B) LL(Z(B)) dim(J 2(Z(B)))

1+2 B0 3 7+ 23 3 3 1+2 b0 3 7+ 23 3 1

5.6 The Rudvalis group2

5.6.1 Ru, p = 3

Block information:

Block B Defect δ(B) k(B) LL(Z(B)) dim(J 2(Z(B)))

1+2 B0 3 3+ 14 3 4 1+2 b0 3 3+ 14 3 3

5.6.2 Ru, p = 5

Block information:

Block B Defect δ(B) k(B) LL(Z(B)) dim(J 2(Z(B)))

1+2 B0 3 5+ 25 3 4 1+2 b0 3 5+ 25 3 3

2 For p = 2, NG(P ) has 85 conjugacy classes. 5.7. THE O’NAN GROUP 115 5.7 The O’Nan group

5.7.1 O0N, p = 2

Block information:

Block B Defect δ(B) k(B) LL(Z(B)) dim(J 2(Z(B)))

B0 9 D 20 LL > 3 3

b0 9 D 32 3 4

∼ 3 where D = Syl2(4 .L3(2)).

5.7.2 O0N, p = 7

Block information:

Block B Defect δ(B) k(B) LL(Z(B)) dim(J 2(Z(B)))

1+2 B0 3 7+ 24 3 3 1+2 b0 3 7+ 24 3 2

5.8 A conjecture

Based on the results presented in this chapter, we make the following conjecture.

Conjecture 5.8.1. Let G be a sporadic simple group and P a Sylow p-subgroup of G; let k be an algebraically closed ﬁeld of characteristic p. Then

LL(Z(B0(kG))) ≥ LL(Z(b0(kNG(P )))).

If P is abelian, the abelian defect group conjecture predicts equality of the Loewy lengths; hence this conjecture is only interesting when P is not abelian. Chapter 6

Blocks with normal, abelian defect groups

The Loewy length of a block has been studied in detail and many structural results have been obtained; for example a Loewy length of 2 or 3 can only appear in a very restricted number of cases [Oku86, Theorem 1]. The outcome is not so clear when considering the centre of a block. In this chapter, we turn our attention to the centres of blocks with normal, elementary abelian defect groups. As usual, k denotes an algebraically closed ﬁeld of characteristic p > 0, and B a block of the group algebra kG of a ﬁnite group G over k. Moreover, D denotes a defect group of B.

6.1 A reduction for the Loewy length calculation

G Let b be a block of DCG(D) such that b = B. The stabiliser of b in NG(D), given by

−1 T := NG(D, b) = {g ∈ NG(D) | g bg = b}

is called the inertial subgroup of b. Note that CG(D) ≤ T ≤ NG(D). Moreover, 0 e(B) = |T : DCG(D)| is a p -number by [Zim14, Section 5.10.4], called the inertial index of B (for more details, also see [Nav98, Chapter 9]). The inertial subgroup T acts on kZ(D) and the algebra of ﬁxed points is denoted by

kZ(D)T := {x ∈ kZ(D) | t−1xt = x ∀t ∈ T }.

116 6.1. A REDUCTION FOR THE LOEWY LENGTH CALCULATION 117

6.1.1 Blocks with cyclic defect groups

In [KKS14], Koshitani, K¨ulshammerand Sambale concentrate on the Loewy length of a block, however for blocks with cyclic defect groups, they also provide an explicit formula for the Loewy length of the centre of a block. We provide a brief summary of the relevant results, which we then generalise as far as possible to blocks with abelian defect groups. The following preliminary result holds for any block B.

Proposition 6.1.1. [KKS14, Corollary 2.7] Let B be a block with maximal Brauer pair (D, b) and inertial subgroup T = NG(D, b). Moreover, let T = T/CG(Z(D)). Then LL(kZ(D)) − 1 LL(B) ≥ LL(Z(B)) ≥ LL(kZ(D)T ) ≥ + 1. | T | The following corollary gives an explicit formula to calculate the Loewy length of the centre of a block with cyclic defect groups.

Corollary 6.1.2. [KKS14, Corollary 2.8] Let B be a p-block with a cyclic defect group D. Then |D| − 1 LL(B) ≥ LL(Z(B)) = + 1 e(B) where e(B) denotes the inertial index of B.

In [KKS14], the proof of Corollary 6.1.2 makes use of the following result by Dade for blocks with cyclic defect groups.

The main step in the proof of Corollary 6.1.2 is the observation that the centre can be decomposed into a direct sum. If H = D o T , with T as in Proposition 6.1.1, then Z(B) ∼= Z(kH) by [K¨ul85,Theorem A]. It then follows by Theorem 6.1.3 that

T Z(kH) = kD ⊕ I1(kH), where I1(kH) is the subspace of Z(kH) spanned by all the class sums of defect 0, an ideal in Z(kH) contained in Soc(kH) (see Lemma 1.6.3). 118 CHAPTER 6. NORMAL ABELIAN DEFECT GROUPS

The aim of the next section is to show that a similar decomposition can be achieved under the weaker assumption that D is abelian; in particular, we concentrate on groups of the form G = D o E where E acts faithfully on D, p - |E| and D is an abelian p-group.

6.1.2 LL(Z(kG)) = LL(kDE)

Throughout this section, we consider groups of the form G = D o E where D is an abelian p-group, p - |E| and E acts faithfully on D. Note that in Sections 6.2 and 6.3, we restrict to D being elementary abelian; results which can be applied more generally are presented here. In order to understand the group algebra, we need to understand the conjugacy classes which occur in these groups.

Lemma 6.1.4. Let G = D o E. Suppose CG(e) ≤ E for all e ∈ E \{1}.

(i) If D is abelian, then CG(d) = D for all d ∈ D \{1}; moreover CG(D) = D.

(ii) For all x ∈ G \ D, xG = eG = D · eE where x = de for some d ∈ D, e ∈ E \{1}. In particular, if E is abelian, then eG = D · e for all e ∈ E \{1}.

Proof. (i) Let d ∈ D \{1}. Then D ≤ CG(D). Suppose g = he ∈ G where h ∈ D, e ∈ E. Then g ∈ CG(d) ⇔ e ∈ CG(d) ⇔ d ∈ CG(e). By assumption, CG(e) ≤ E.

Hence e = 1 and CG(d) = D. Finally, CG(D) = ∩d∈DCG(d) = ∩D = D. (ii) First consider e ∈ E \{1}, and let g = dh ∈ G where d ∈ D, h ∈ E. Then eg = edh = h−1(d−1ed)h = h−1(d−1d∗e)h = (d−1d∗)heh ∈ DeE. On the other hand,

|G| |G| |eG| = = = |eE| · |D|. |CG(e)| |CE(e)| Thus eG = D · eE. Let x ∈ G \ D. Then there exists e ∈ E \{1} such that x ∈ De. Hence xG ⊆ DeG = DeE = eG. Therefore xG = eG = DeE. Obviously, if E is abelian, then eE = E and hence eG = De for all e ∈ E \{1}.

Now that the conjugacy classes of G = D o E are understood, Lemma 6.1.4 can be used to decompose the centre of the group algebra. As we will see in Theorem 6.1.7, one summand is contained in the socle of kG, and hence this decomposition will give the required reduction. 6.1. A REDUCTION FOR THE LOEWY LENGTH CALCULATION 119

For groups G = D o E, let kDE denote the algebra of ﬁxed points of kD under the action of E, i.e. kDE = {x ∈ kD | e−1xe = x ∀e ∈ E}.

Theorem 6.1.5. Let G = DoE where D is an abelian p-group, and assume CG(D) = E E D. Then kD = kD ∩ Z(kG) and Z(kG) = kD ⊕ ker(BrD).

Proof. We ﬁrst show kDE = kD ∩ Z(kG).

As G = D o E, then for g ∈ G, there exist e ∈ E, d ∈ D such that g = de. Hence, as D is abelian, g−1xg = e−1d−1xde = e−1xe for all x ∈ D. Thus

kDE = {x ∈ kD | e−1xe = x ∀e ∈ E} = {x ∈ kD | g−1xg = x ∀g ∈ G} = kDG.

Therefore, kDE = kDG ⊆ Z(kG) ∩ kD. On the other hand, we clearly have Z(kG) ∩ kD ⊆ kDE

Next we note that, since by assumption CG(D) = D,

ker(BrD) = hCb(x) | C (x) ∩ CG(D) = ∅ik

= hCb(x) | C (x) ∩ D = ∅ik P = { βxx | βx ∈ k, βxg = βx ∀g ∈ G} x∈G\D

E P Moreover, kD = kD ∩ Z(kG) = { αxx | αx ∈ k, αxg = αx ∀g ∈ G}. Thus x∈D

P Z(kG) = { γxx | γx ∈ k, γxg = γx ∀g ∈ G} x∈G P P = { γxx + γxx | γx ∈ k, γxg = γx ∀g ∈ G} x∈D x∈G\D E = kD + ker(BrD).

E Hence it remains to show that kD ∩ ker(BrD) = 0. E P P Suppose r ∈ kD ∩ ker(BrD). Then r = αxx = βxx. Now since D is a x∈D x∈G\D normal subgroup, for all x ∈ G \ D, C (x) ∩ D = ∅. Hence, αx = βx = 0 for all x ∈ G, E and so r = 0. Therefore Z(kG) = kD ⊕ ker(BrD).

Recall from Deﬁnition 1.4.3 that I1(kG) denotes the subspace of Z(kG) spanned by all class sums of defect zero.

Lemma 6.1.6. Let G be a ﬁnite group and D a non-trivial p-subgroup of G. Then

I1(kG) is contained in ker(BrD). 120 CHAPTER 6. NORMAL ABELIAN DEFECT GROUPS

Proof. Let C (x) ∈ I1(kG). Then |CG(x)|p = 1 and hence D *G Dx = {1} where

Dx ∈ δ(C (x)) is a defect group of C (x). Therefore, by Theorem 1.4.2,

X kCb(x) ⊆ ker(BrD).

x∈G, Dx={1}

The next theorem forms the main result of this section, allowing us to reduce to the conjugacy classes of elements in D.

Theorem 6.1.7. Let G = D oE where D is an abelian p-group and p - |E|. Suppose

CG(e) ≤ E for all e ∈ E \{1}. Then

ker(BrD) = I1(kG) ⊆ Soc(kG) and hence LL(Z(kG)) = LL(kDE).

Proof. Let x ∈ G \ D. Then x = de for some d ∈ D, e ∈ E \{1}. Moreover, by

G G E Lemma 6.1.4(i), x = e = De . Now let Cb(x) ∈ ker(BrD). Then C (x) ∩ D = ∅ and

|C (x)| = (|E|/|CG(e)|) · |D| = (|E|/|CE(e)|) · |D|. Hence |C(x)|p = |D|p = |G|p and so

Cb(x) ∈ I1(kG). Thus ker(BrD) ⊆ I1(kG) and so by Lemma 6.1.6, I1(kG) = ker(BrD).

E By Lemma 6.1.4(i), CG(D) = D and hence by Theorem 6.1.5, Z(kG) = kD ⊕ ker(BrD)}. Finally, by Lemma 1.6.3, ker(BrD) = I1(kG) ⊆ Soc(kG), and we can conclude that LL(Z(kG)) = LL(kDE).

With the establishment of our main reduction, we concentrate on calculating the Loewy length of the centre for speciﬁc examples. A block has Loewy length two if and only if D is isomorphic to C2:

Theorem 6.1.8. [KKS14, Proposition 3.1] Let B be a p-block with defect d(B). Then

LL(B) = 2 ⇔ p = 2 and d(B) = 1.

The situation is very diﬀerent when considering the centre of a block. 6.2. CALCULATING SOME EXAMPLES 121

Theorem 6.1.9. Let G = D oE where D is an abelian p-group and p - |E|. Assume G CG(e) ⊆ E for all e ∈ E \{1} and suppose E acts transitively on D, i.e. d = D\{1} for all d ∈ D\{1}. Then LL(Z(kG)) = 2.

Proof. By Theorem 6.1.7, LL(Z(kG)) = LL(kDE). Since the action is transitive, we E P E have kD = Z(kG) ∩ kD = {1, x∈D\{1} x } and by Proposition 1.5.10, J(kD ) = P spank{ x∈D\{1} x − (|D| − 1)} = spank{Db}. Furthermore

DbDb = |D|Db ≡ 0 mod J(O)G giving the result.

Question 6.1.10. Is the reverse implication of Theorem 6.1.9 also true? As we will see in the next section, examples seem to suggest that for groups as considered in Theorem 6.1.9, a transitive action of E on D is a necessary and suﬃcient condition to obtain LL(Z(kG)) = 2.

Note that since E acts transitively on D, exp(D) = p. Therefore, as D is abelian and exp(D) = p, the subgroup D is elementary abelian of order pn for some n. With this is mind, we shall study the cases where D is an elementary abelian p-group.

6.2 Calculating some examples

The aim of this chapter is to consider what happens when the condition of a cyclic defect group in Lemma 6.1.2 is replaced by an elementary abelian defect group. From ∼ n now on, the focus will be on groups of the form G = D o E where D = (Cp) and

E ≤ Aut(D) such that CG(e) ≤ E for all e ∈ E \{1}. In this section, some small examples are presented which will enable us to establish some patterns in LL(Z(kG)); these are then proven in Section 6.3.

Consider D = C5 × C5 = hai × hbi. We can take the automorphism e of order e e 3 24 such that a = ab and b = a b. It follows that C (a) = (C5 × C5) \{1} and thus

CG(a) = D; moreover, CG(d) = D for all d ∈ D \{1}. 122 CHAPTER 6. NORMAL ABELIAN DEFECT GROUPS

Remark 6.2.1. Note that this can be done in general for an elementary abelian group n ∼ D = (Cp) . In particular, there exists E = Cpn−1 ≤ Aut(D) such that for G = D o E, dG = D \{1} (this is the Singer cycle). Suppose E = hei. Here we consider subgroups of the form D o heii for i | pn − 1.

Further, note that all groups considered are p-solvable, with Op0 (G) = 1. Hence by Proposition 1.4.15, the group algebra kG is indecomposable.

We now run through some examples; the computations of LL(Z(kG)) were carried out in [GAP]. All the results are summarised in a table at the end of this section, which includes the small groups ID for each group considered.

Let D = C5 × C5 and start with (C5 × C5) o C24 where C24 acts faithfully and transitively on D. Take the following generators for G:

5 5 24 −1 −1 3 G = (C5 × C5) o C24 = ha, b, c | a = b = c , ab = ba, c ac = ab, c bc = a bi

There are 25 conjugacy classes

C (g) 1A 5A 1 ≤ i ≤ 23 |C (g)| 1 24 25

|CG(g)| 600 25 24 i elements 1G D \{1} Dc and J(Z(kG)) = {D,ˆ Dcˆ i}. Since D is a normal subgroup, DˆDˆ = |D|Dˆ ≡ 0 mod J(O)G, and Dcˆ iDcˆ j = |D|Dcˆ icj ≡ 0 mod J(O)G for all 1 ≤ i, j ≤ 23. Hence LL(Z(kG))=2.

(C5 × C5) o C12

5 5 12 −1 4 2 −1 4 G = (C5 × C5) o C12 = ha, b, c | a = b = c , ab = ba, c ac = a b , c bc = ab i

C (g) 1A 2A 3A 3B 4A 4B 5A 5B 6A 6B 12A 12B 12C 12D |C (g)| 1 25 25 25 25 25 12 12 25 25 25 25 25 25

|CG(g)| 300 12 12 12 12 12 25 25 12 12 12 12 12 12 6 4 8 3 9 G G 2 10 5 7 11 elements 1G Dc Dc Dc Dc Dc a b Dc Dc Dc Dc Dc Dc

Here, J(Z(kG)) = {Cb(a) + 3, Cb(b) + 3, Dcˆ i}.

• Dcˆ iDcˆ j = |D|Dcˆ icj ≡ 0 mod J(O)G for all 1 ≤ i, j ≤ 11

•∀ x ∈ {5A, 5B}, ∀y 6∈ {1A, 5A, 5B},(Cb(x) + 3)Cb(y) = (12Dˆ + 3Dˆ)ci ≡ 0 6.2. CALCULATING SOME EXAMPLES 123

• (Cb(x) + 3)2 = 1 + Cb(5A) + Cb(5B) = Dˆ

• (Cb(x) + 3)3 = 0

LL(Z(kG))=3

(C5 × C5) o C8

5 5 8 −1 −1 3 G = (C5 × C5) o C8 = ha, b, c | a = b = c , ab = ba, c ac = b, c bc = a i

C (g) 1A 2A 4A 4B 5A 5B 5C 8A 8B 8C 8D |C (g)| 1 25 25 25 8 8 8 25 25 25 25

|CG(g)| 200 8 8 8 25 25 25 8 8 8 8 4 2 6 G G 4 2 G 3 5 7 elements 1G Dc Dc Dc a (ab) (a b ) Dc Dc Dc Dc

Here, J(Z(kG)) = {Cb(a) + 2, Cb(ab) + 2, Cb(a4b2) + 2, Dcˆ i}.

•∀ x ∈ {5A, 5B, 5C}, ∀y 6∈ {1A, 5A, 5B, 5C},(Cb(x) + 2)Cb(y) ≡ 0 mod J(O)G

• (Cb(5A) + 2)2 = 2 + 2Cb(5A) + 2Cb(5B) + 2Cb(5C) = 2Dˆ

• (Cb(5A) + 2)3 = 0

LL(Z(kG))=3 124 CHAPTER 6. NORMAL ABELIAN DEFECT GROUPS

The following table summarises the Loewy lengths calculated here, and in GAP. Where available, the given Group ID refers to the Small Group Library in [GAP].

Group Group ID LL(Z(kG)) Group Group ID LL(Z(kG))

4 (C2 × C2) o C3 [12,3] 2 (C2) o C3 [48,50] 3

3 4 (C2) o C7 [56,11] 2 (C2) o C5 [80,49] 3

5 4 (C2) o C31 [992,194] 2 (C2) o C15 [240,191] 2

(C3 × C3) o C2 [18,4] 3 (C3 × C3 × C3) o C2 [54,14] 4

(C3 × C3) o C4 [36,9] 3 (C3 × C3 × C3) o C13 [351,12] 3

(C3 × C3) o C8 [72,39] 2 (C3 × C3 × C3) o C26 [702,48] 2

(C5 × C5) o C2 [50,4] 5 (C5 × C5 × C5) o C2 [250,14] 7

(C5 × C5) o C3 [75,2] 5 (C5 × C5 × C5) o C4 [500,48] 4

(C5 × C5) o C4 [100,11] 3 (C5 × C5 × C5) o C31 [3875,16] 5

(C5 × C5) o C6 [150,6] 5

(C5 × C5) o C8 [200,40] 3

(C5 × C5) o C12 [300,24] 3 (C5 × C5 × C5) o C62 3

(C5 × C5) o C24 [600,149] 2 (C5 × C5 × C5) o C124 2

(C7 × C7) o C2 [98,4] 7 (C7 × C7 × C7) o C2 [686,14] 10

(C7 × C7) o C3 [147,4] 5

(C7 × C7) o C4 [196,8] 7

(C7 × C7) o C6 [294,13] 3

(C7 × C7) o C8 [392,36] 7

(C7 × C7) o C12 [588,34] 3

(C7 × C7) o C16 [784,160] 4

(C7 × C7) o C24 [1176,213] 3

(C7 × C7) o C48 2 6.2. CALCULATING SOME EXAMPLES 125

Group Group ID LL(Z(kG))

(C11 × C11) o C2 [242,4] 11

(C11 × C11) o C3 [363,2] 11

(C11 × C11) o C4 [484,8] 11

(C11 × C11) o C5 [605,4] 5

((C11 × C11) o C3) o C2 [726,6] 11

((C11 × C11) o C5) o C2 [1210,9] 3

((C11 × C11) o C4) o C3 [1452,21] 11

((C11 × C11) o C5) o C3 [1815,3] 5

((C11 × C11) o C5) o C4 [2420,44] 3

(((C11 × C11) o C5) o C3) o C2 [3630,15] 3

(C11 × C11) o C60 [7260,93] 3

(C11 × C11) o C120 2

From the examples considered, the following observations can be made, where the actions in the semidirect product are taken as a subgroup of the Singer cycle, see Remark 6.2.1.

n • If G = (Cp) o Cpn−1 then LL(Z(kG)) = 2.

n • If p is odd and G = (Cp) o C pn−1 then LL(Z(kG)) = 3. 2

n p−1 • If p is odd and G = (Cp) o C2 then LL(Z(kG)) = n × 2 + 1.

n • If p is odd and G = (Cp) o Cp−1 then LL(Z(kG)) = n + 1.

Note that for n = 1, these formulae agree with the result in Corollary 6.1.2. Moreover, the ﬁrst observation has already been proven in Theorem 6.1.9, while we will prove the other three in the next section. 126 CHAPTER 6. NORMAL ABELIAN DEFECT GROUPS

n 6.3 General formulae for the Loewy length of (Cp) o E

Throughout this section, assume p > 2. Furthermore, we only consider groups of the form G = D o E where E ≤ Aut(D), p - |E|, and the action of E on D is faithful.

Moreover, CG(e) ≤ E for all e ∈ E \{1}, and by Lemma 6.1.4(i), CG(D) = D.

n As we have seen in Theorem 6.1.9, if E acts transitively on D = (Cp) then LL(Z(kG)) = 2. Thus it is natural to ask what happens when a subgroup of E acts on D. The ﬁrst such action we consider is when the action splits the elements of

n pn−1 D = (Cp) into two orbits of length 2 .

Theorem 6.3.1. Let G = D o E where D is elementary abelian of order pn and ∼ pn−1 E = C pn−1 , where the action of E on D splits D into two orbits of length . Then 2 2

LL(Z(kG)) = 3.

Proof. We have CG(e) ⊆ E for all e ∈ E \{1}, hence by Theorem 6.1.7, LL(Z(kG)) = LL(kDE). Due to the action of E on D, it follows by Theorem 6.1.5 that kDE = Z(kG) ∩ kD = {1, Cb(x), Cb(y)}, where x, y ∈ D \{1} such that y 6∈ C (x). Thus E pn−1 pn−1 J(kD ) = spank{Cb(x) − 2 , Cb(y) − 2 } by Proposition 1.5.10. pn+3 By Lemma 6.1.4(ii), the group G has 2 conjugacy classes, which are of the pn+3 form C (1), C (x), C (y), {De | e ∈ E \{1}}. Thus G has 2 irreducible characters. ∼ pn−1 As G/D = E, G has at least 2 linear characters. By [Isa94, Theorem 6.34], if G G pn−1 θ ∈ Irr(D) is non-trivial, then θ ∈ Irr(G); thus θ has degree |G : D|θ(1) = 2 . pn−1 Hence we obtain two more irreducible characters of degree 2 in G. Hence the pn−1 character table for G has the following generic form, with 2 linear characters:

1 x y p0 − elements 1 1 1 1 . . 1 1 ? 1 1 1 ? (pn − 1)/2 α β 0 (pn − 1)/2 β α 0

From the orthogonality relations, Theorem 1.2.2, it is clear that β = −1−α; moreover there are two cases depending on whether an element x in D is conjugate to its inverse N 6.3. GENERAL FORMULAE FOR THE LOEWY LENGTH OF (CP ) o E 127 or not: √ −1 G −1± pn x ∈ x ⇒ α, β = 2 √ −1 G −1± −pn x 6∈ x ⇒ α, β = 2 √ √ −1 G −1+ pn −1− pn Case 1: Suppose x ∈ x for all x ∈ D; hence α = 2 , β = 2 . Let x, y ∈ D \{1}, y 6∈ C (x). Burnside’s formula, Equation 1.2.2, yields the following equivalences modulo J(O)G:

a(x, y, 1) = 0 pn−1 a(x, y, x) ≡ a(x, x, y) ≡ a(x, y, y) ≡ 4 pn−1 a(x, x, 1) ≡ 2 pn−5 a(x, x, x) ≡ 4

Hence the elements of J(kDE) multiply as follows:

2 pn−1 2 n pn−1 2 Cb(x) − 2 = Cb(x) − (p − 1)Cb(x) + 2

pn−5 4 pn−1 p2n−1 = 4 + 4 Cb(x) + 4 Cb(y) + 4 1G 1 ≡ − 4 Db mod J(O)G

2 pn−1 1 Similarly, Cb(y) − 2 ≡ − 4 Db mod J(O)G.

pn−1 pn−1 (pn−1) (pn−1) Cb(x) − 2 Cb(y) − 2 = Cb(x)Cb(y) − 2 Cb(x) − 2 Cb(y)

p2n−2pn+1 + 4 1G

−pn+1 −pn+1 p2n−2pn+1 = 4 Cb(x) + 4 Cb(y) + 4 1G 1 ≡ 4 Db mod J(O)G

√ √ −1 G −1+ −pn −1− −pn Case 2: Suppose x 6∈ x for all x ∈ D \{1}; hence α = 2 , β = 2 . Let x, y ∈ D \{1}, y 6∈ C (x). Using Burnside’s formula, the following equivalences modulo J(O)G are obtained:

pn−1 a(x, y, 1) ≡ 2 pn−3 a(x, y, x) ≡ a(x, y, y) ≡ 4 a(x, x, 1) = 0 pn−3 a(x, x, x) ≡ 4 pn+1 a(x, x, y) ≡ 4 128 CHAPTER 6. NORMAL ABELIAN DEFECT GROUPS

Hence the elements of J(kDE) multiply as follows:

2 pn−1 2 n pn−1 2 Cb(x) − 2 = Cb(x) − (p − 1)Cb(x) + 2

pn−3 4 pn+1 p2n−2pn+1 = 0 + 4 + 4 Cb(x) + 4 Cb(y) + 4 1G 1 ≡ 4 Db mod J(O)G

2 pn−1 1 Similarly, Cb(x) − 2 ≡ 4 Db mod J(O)G.

pn−1 pn−1 (pn−1) (pn−1) Cb(x) − 2 Cb(y) − 2 = Cb(x)Cb(y) − 2 Cb(x) − 2 Cb(y)

p2n−2pn+1 + 4 1G

−pn−1 −pn−1 p2n−1 = 4 Cb(x) + 4 Cb(y) + 4 1G 1 ≡ − 4 Db mod J(O)G

pn−1 ˆ pn−1 ˆ pn−1 ˆ pn−1 ˆ Finally, since (Cb(x) − 2 )D = (Cb(y) − 2 )D = 2 D − 2 D = 0, LL(kDE) = 3.

n ∼ The next aim is to consider another two speciﬁc actions on (Cp) = hx1i × ... × hxni; pn−1 the ﬁrst one splits the non-trivial elements of D into 2 conjugacy classes of the form −1 pn−1 C (d) = {d, d } for all d ∈ D \{1} and the second splits them into p−1 conjugacy j classes of the form C (xi) = {xi | 1 ≤ j < p}. The following lemma is required to establish a lower bound on LL(kDE).

Lemma 6.3.2. Let g ∈ Cp \{1}, and k a ﬁeld of characteristic p. Then there exists

α ∈ kCp such that

p−1 Y i p−1 α · (g − 1) = (1 + g + ... + g ) ∈ kCp. i=1

Proof. Let g ∈ Cp. We start by claiming that there exist elements αi ∈ kCP such that i αi(g − 1) = (g − 1) for all 1 ≤ i ≤ p − 1. Let k ∈ N. Then for all 1 ≤ i ≤ p − 1,

(1 + gi + g2i + ... + gki) · (gi − 1) = g(k+1)i − 1.

(ki+1)i i ki+1 i We can ﬁx ki ∈ N such that g = (g ) = g. Hence given (g − 1), there exists i αi ∈ kCp such that αi(g − 1) = g − 1. N 6.3. GENERAL FORMULAE FOR THE LOEWY LENGTH OF (CP ) o E 129

Hence there exists α ∈ kCp such that

p−1 p−1 Y X p − 1 α · (gi − 1) = (g − 1)p−1 = (−1)i g(p−1)−i i i=1 i=0 where the last equality is a simple application of the binomial formula. Each coeﬃcient is given by the form

ip−1 i (p−1)...(p−1−i+1) (−1) i = (−1) i(i−1)...2 i (p−1)...(p−i) = (−1) i(i−1)...2 i (p−i) (p−(i−1)) (p−2) (p−1) = (−1) · i · i−1 · ... · 2 · 1 ≡ (−1)i · (−1)i = 1

Qp−1 i Pp−1 i where the equivalence is taken modulo p. Thus α · i=1 (g − 1) ≡ i=0 g mod p.

n ∼ Theorem 6.3.3. Let G = D o E where D is elementary abelian of order p , E = C2, and for d ∈ D, e ∈ E \{1}, we have de = d−1. Then p − 1 LL(Z(kG)) = n × + 1. 2

Proof. We have CG(e) ⊆ E for all e ∈ E\{1}, and hence by Theorem 6.1.7, LL(Z(kG)) = E pn−1 LL(kD ). The action of E divides D into 2 conjugacy classes of size 2, and the E −1 identity; by Proposition 1.5.10, J(kD ) = spank{x + x − 2 | x ∈ D \{1}}. We split this proof into two parts, proving the following two inequalities.

p−1 (I) LL(Z(kG)) ≤ n × 2 + 1 p−1 (II) n × 2 < LL(Z(kG))

n Part (I): As D = (Cp) is abelian, by Theorem 1.5.8,

LL(Z(kD)) = LL(kD) = n × p − (n − 1) = n(p − 1) + 1 =: 2s − 1 (s ∈ N).

Hence any 2s − 1 elements in J(kD) multiply to zero modulo J(O)G; for example

−1 −1 (x1 − 1)(x1 − 1)(x2 − 1)(x2 − 1) ... (xs − 1) ≡ 0 for all xi ∈ D, not necessarily distinct. Next consider an arbitrary multiplication of s elements in J(kDE); by above, these

−1 −1 are of the form (xi + xi − 2) = (xi − 1)(xi − 1)(−1). Hence s s Y −1 Y −1 (xi + xi − 2) = (xi − 1)(xi − 1)(−1) ≡ 0 mod J(O)G. i=1 i=1 130 CHAPTER 6. NORMAL ABELIAN DEFECT GROUPS

E p−1 and so LL(kD ) ≤ s = n × 2 + 1. p−1 E Part (II): It is enough to ﬁnd n × 2 elements in J(kD ) whose product is not congruent to zero. Consider g ∈ Cp. Then

p−1 p−1 p−1 2 2 2 p−1 Y i −i p−1 Y i Y −i p−1 Y i (g + g − 2) = (−1) 2 (g − 1) (g − 1) = (−1) 2 (g − 1). i=1 i=1 i=1 i=1

p−1 2 Hence by Lemma 6.3.2, there exists an element α ∈ kD such that α· Q (gi +g−i −2) ≡ i=1 p−1 P i g . Thus when considering all n copies of Cp inside D, there exists α ∈ kD such i=0 that p−1 p−1 n 2 n p−1 n 2 Q Q i −i Q P i ˆ Q Q i −i α· (gj +gj −2) ≡ gj = D. In particular, (gj +gj −2) 6≡ 0. j=1 i=1 j=1 i=0 j=1 i=1

2 ∼ Corollary 6.3.4. Let G = (Cp) oE with E = C2 such that for all d ∈ D, e ∈ E \{1}, we have de = d−1. Then LL(Z(kG)) = p.

Proof. This is the case n = 2 in Theorem 6.3.3.

Theorem 6.3.5. Let G = D o E where D is elementary abelian of order pn and ∼ E = Cp−1, where the action of E on D is such that for all x ∈ D\{1}, C (x) = hxi\{1}. Then

LL(Z(kG)) = n + 1.

Proof. We have CG(e) ⊆ E for all e ∈ E\{1}, and hence by Theorem 6.1.7, LL(Z(kG)) = LL(kDE). Under the action of E, the non-trivial elements of the normal subgroup pn−1 D split into p−1 conjugacy classes of size p − 1; by Proposition 1.5.10, and since E −|Cb(x)| = −(p − 1) ≡ +1 mod J(O)G, J(kD ) = spank{Cb(x) + 1 | x ∈ kD} = 2 p−1 spank{(x + x + ... + x + 1) | x ∈ kD}. The proof will be split into two parts as follows:

(I) LL(Z(kG)) ≤ n + 1 (II) n < LL(Z(kG))

n Part (I): As D = (Cp) is abelian, by Theorem 1.5.8,

LL(Z(kD)) = LL(kD) = n × p − (n − 1) = n(p − 1) + 1. N 6.3. GENERAL FORMULAE FOR THE LOEWY LENGTH OF (CP ) o E 131

Set s := n+1 and consider an arbitrary multiplication of s elements in J(kDE). Then, by Lemma 6.3.2, there exists α ∈ kD such that

2 3 p−1 2 3 p−1 (1 + x1 + x + x + ... + x ) ... (1 + xs + x + x + ... + x ) 1 1 1 s s s (6.3.1) 2 p−1 2 p−1 = α[(x1 − 1)(x1 − 1) ... (x1 − 1)] ... [(xs − 1)(xs − 1) ... (xs − 1)]

The second equality contains s(p − 1) factors and since s(p − 1) = (n + 1)(p − 1) = n(p − 1) + (p − 1) ≥ n(p − 1) + 1 = LL(Z(kD)) (p ≥ 2), it follows that both sides of Equation 6.3.1 are zero.

n Part (II): Let xi generate the ith component of the direct product D = (Cp) . Then the set ˆ 2 3 p−1 {Di = (xi + xi + xi + ... + xi + 1) | 1 ≤ i ≤ n}

E Qn ˆ ˆ contains n distinct elements of J(kD ). Clearly, i=1 Di = D 6≡ 0 mod J(O)G. This completes the proof of the theorem.

We ﬁnish this section with two conjectures, which are based on the calculations made in Section 6.2. Note that these only hold if D is elementary abelian of order p2 and cannot be extended to an arbitrary elementary abelian p-group.

Conjecture 6.3.6. Suppose G := (Cp × Cp) o E. Then

∼ LL(Z(kG)) = 3 ⇔ (Cp × Cp) o Cp−1 ¢ G and E 6= C|D|−1.

Conjecture 6.3.7. Suppose G := (Cp × Cp) o E. Then

LL(Z(kG)) = p ⇔ (Cp × Cp) o Cp+1 ¤ G ¤ (Cp × Cp) o C2. Bibliography

[Alp86] J.L. Alperin, Local representation theory, Cambridge University Press, Cam- bridge (1986).

[AE05] J. An and C.W. Eaton, Modular representation theory of blocks with trivial intersection defect groups, Algebr. Represent. Theory 8 (2005), 427-448.

[Ben95] D.J. Benson, Representations and cohomology I, Cambridge University Press, Cambridge (1995).

[BM90] H. Blau and G. Michler, Modular representation theory of ﬁnite groups with T.I. Sylow p-subgroups, Trans. Amer. Math. Soc 319 (1990), 417-468.

[BZ15] S. Bouc and A. Zimmermann, On a question of Rickard on tensor products of stably equivalent algebras, to appear in: Experimental Mathemmatics, preprint (2015).

[Bra56] R. Brauer, Zur Darstellungstheorie der Gruppen endlicher Ordnung, Math.Z. 63 (1956) 406-444.

[Bra63] R. Brauer, Representations of ﬁnite groups, In: Lectures on Modern Mathe- matics 1, Wiley New York (1963), 133-175.

[BN41] R. Brauer and C. Nesbitt, On the modular characters of groups, Ann. of Math. (2) 42 (1941), 556-590.

[Bro90] M. Broué, Isométriesparfaites, types de blocs, catégoriesdérivées, Astérisque 181-182 (1990), 61-92.

[Bro94] M. Brou´e, Equivalences of blocks of group algebras, In: Finite-dimensional

132 BIBLIOGRAPHY 133

algebras and related topics (Ottawa, 1992), (Eds. V.Dlab, L.L.Scott), Kluwer Acad. Publ., Dordrecht (1994), 1-26.

[Bro15] J. Brough, Recognizing abelian Sylow subgroups in ﬁnite groups, Comm. Al- gebra 43 (2015), 5347-5361.

[Bur11] W. Burnside, Theory of groups of ﬁnite order, 2nd edn. (Cambridge University Press, Cambridge (1911)), reprinted (Dover, New York, (1955)).

[Car89] R.W. Carter, Simple groups of Lie type, Wiley Classics Library, John Wiley and Sons, New York (1989).

[Cli00] G. Cliﬀ, On centers of 2-blocks of Suzuki groups, J. Algebra 226 (2000), 74-90.

[CCN+85] J.H. Conway, R.T. Curtis, S.P. Norton, R.A. Parker, and R.A. Wilson, Atlas of fnite groups, Oxford University Press, Eynsham (1985).

[CR62] C.W. Curtis and I. Reiner, Representation theory of ﬁnite groups and associa- tive algebras, John Wiley and Sons, New York-London (1962).

[CR81] C.W. Curtis and I. Reiner, Methods of Representation Theory (2 vols.), John Wiley and Sons, New York (1981).

[Dad66] E.C. Dade, Blocks with cyclic defect groups, Ann. of Math. (2) 84 (1966), 20-48.

[Eat00] C.W. Eaton, Dade’s inductive conjecture for the Ree groups of type G2 in the deﬁning charactersitic, J. Algebra 226 (2000), 614-620.

[Eat01] C.W. Eaton, On ﬁnite groups of p-local rank one and conjectures of Dade and Robinson, J. Algebra 238 (2001), 623-642.

[EK08] S. Ensslen and B. K¨ulshammer, A note on blocks with dihedral defect groups, Arch. Math. 91 (2008), 205-211.

[FK07] Y. Fan and B. K¨ulshammer, A note on blocks with abelian defect groups, J. Algebra 317 (2007), 250-259.

[Fon61] P. Fong, On the characters of p-solvable groups, Trans.Amer.Math.Soc 98 (1961), 263-284. 134 BIBLIOGRAPHY

[Fon] P. Fong, Isotypies and Shintani theory in SL(2, q), unpublished manuscript.

[GAP] The GAP Group, GAP- Groups, Algorithms and Programming, Version 4.7.7 (2015), http://www.gap-system.org.

[Gor80] D. Gorenstein, Finite groups, 2nd edn, Chelsea Publishing Company, New York (1980).

[GL83] D. Gorenstein and R. Lyons, The local structure of ﬁnite grous of characteristic 2-type, Mem.Amer.Math.Soc 42 (1983).

[Gra05] J.B. Gramain, Generalized Block Theory, PhD Thesis (2005).

[Gre68] J.A. Green, Some remarks on defect groups, Math.Z. 107 (1968), 133-150.

[HKK10] M. Holloway, S. Koshitani and N. Kunugi, Blocks with nonabelian defect groups which have cyclic subgroups of index p, Arch. Math. 94 (2010), 101-116.

[Isa94] I.M. Isaacs, Character theory of ﬁnite groups, Dover Publications (1994).

[JL01] G. James and M. Liebeck, Representations and characters of groups, 2nd Ed, Cambridge University Press (2001).

[Jon94] G.A. Jones, Ree Groups and Riemann Surfaces, J. Algebra 165 (1994), 41-62.

[Kar87] G. Karpilovsky, The Jacobson radical of group algebras, North-Holland Pub- lishing (1987).

[Kar92] G. Karpilovsky, Group representations, Vol. 1, Part B, North-Holland Pub- lishing (1992).

[KZ98] S. K¨onigand A. Zimmermann, Derived equivalences for group rings, Springer- Verlag, Berlin (1998).

[Kos14] S. Koshitani, The projective cover of the trivial module over a group algebra of a ﬁnite group, Comm. Algebra 42 (2014) 4308-4321.

[KKS14] S. Koshitani, B. K¨ulshammerand B. Sambale, On Loewy lengths of blocks, Math. Proc. Cambridge Philos. Soc. 156 (2014), 555-570. BIBLIOGRAPHY 135

[KK02] S. Koshitani and N. Kunugi, Brou´e’sconjecture holds for principal 3-blocks with elementary abelian defect grous of order 9, J. Algebra, 248 (2002), 575-604.

[KM01] S. Koshitani and H. Miyachi, Donovan conjecture and Loewy length for principal 3-blocks of ﬁnite groups with elementary abelian Sylow 3-subgroup of order 9, Comm. Algebra 29 (2001) 4509-4522.

[Kül81] B. Külshammer, Bemerkungen über die Gruppenalgebra als symmetrische Al- gebra, J. Algebra 72 (1981), 1-7.

[Kül82] B. Külshammer, Bemerkungen über die Gruppenalgebra als symmetrische Al- gebra II, J. Algebra 75 (1982), 59-69.

[K¨ul85] B. K¨ulshammer, Crossed products and blocks with normal defect groups, Comm. Algebra 13 (1985), 147-168.

[K¨ul91] B. K¨ulshammer, Group-theoretical descriptions of ring-theoretical invariants of group algebras, Progr. Math. 95 (1991), 425-442.

[Lin91] M. Linckelmann, Derived equivalence for cyclic blocks over a P -adic ring, Math. Z. 207 (1991), 293-304.

[LP10] K. Lux and H. Pahlings, Representations of Groups: A combinatorial approach, Cambridge University Press, Cambridge (2010).

[Mey06] H. Meyer, On a subalgebra of the centre of a group ring, J.Algebra 295 (2006), 293-302.

[NT89] H. Nagao and Y. Tsushima, Representations of ﬁnite groups, Academic Press, Boston (1989).

[Nav98] G. Navarro, Characters and Blocks of Finite Groups, Vol. 250 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cam- bridge (1998).

[NST15] G. Navarro, R. Solomon and P.H. Tiep, Abelian Sylow subgroups in a ﬁnite group, II, J. Algebra 421, (2015), 3-11. 136 BIBLIOGRAPHY

[Oku80] T. Okuyama, Some studies on group algebras, Hokkaido Math. J. 9 (1980), 217-221.

[Oku81] T. Okuyama, On the radical of the center of a group algebra, Hokkaido Math. J. 10 (1981), 406-408.

[Oku86] T. Okuyama, On blocks of ﬁnte groups with radical cube zero, Osaka J. Math. 23 (1986), 461-465.

[Oku98] T. Okuyama, Some examples of derived equivalent blocks of ﬁnte groups, preprint (1998).

[Oku00] T. Okuyama, Derived equivalences in SL(2,q), preprint (2000).

[Pas80] D.S. Passman, The radical of the center of a group algebra, Proc.Amer.Math.Soc 78 (1980), 323-326.

[Ree61] R. Ree, A familiy of simple groups associated with the simpe Lie algebra of

type (G2), Amer.J.Math. 83 (1961), 432-462.

[Rey63] W.F. Reynolds, Blocks and normal subgroups of ﬁnite groups, Nagoya Math. J. 22 (1963), 15-32.

[Rey72] W.F. Reynolds, Sections and ideals of centers of group algebra, J. Algebra 20 (1972), 176-181.

[Ric89] J. Rickard, Derived categories and stable equivalence, J. Pure Appl. Algebra 61 (1989), 303-317.

[Ric91] J. Rickard, Derived equivalences as derived functors, J. London Math.Soc. 43 (1991), 37-48.

[Ric98] J. Rickard, Some recent advances in modular representation theory, Algebras and modules (Trondheim, 1996), 157-178, CMS Conf. Proc. 23, AMS (1998).

[Rob08] D. Robbins, Brou´e’s abelian defect group conjecture for the Tits group, preprint, arXiv:0807.3105v1, (2008).

[Rob84] G.R. Robinson, Some uses of class algebra constants, J. Algebra 91 (1984), 64-74. BIBLIOGRAPHY 137

[Rob96] G.R. Robinson, Local structure, vertices and Alperin’s conjecture, Proc. Lon- don Math. Soc.(3) 72 (1996), 312-330.

[Rob00] G.R. Robinson, A note on perfect isometries, J. Algebra 226 (2000), 71-73.

[Rou98] R. Rouquier, The derived category of blocks with cyclic defect groups, In: Derived equivalences for group rings, Lecture Notes in Math. 1685, Springer, Berlin (1998), 199-220.

[Suz60] M. Suzuki, A new type of simple groups of ﬁnite order, Proc.Nat.Acad.Sci. U.S.A. 46 (1960), 868-870.

[Suz62] M. Suzuki, On a class of doubly transitive groups, Ann. of Math. 75 (1962), 105-145.

[War66] H.N. Ward, On Ree’s series of simple groups, Trans. Amer. Math. Soc, 121 (1966), 62-89.

[Zim14] A. Zimmermann, Representation Theory: A Homological Algebra Point of View, Springer Verlag, London (2014). Appendix A

GAP Code

The following code deﬁning the Loewy length as a function in GAP was kindly provided by Benjamin Sambale.

LoewyLength:=function(A) local J,JJ,ll; if Dimension(A)=0 then return 0; fi; J:=RadicalOfAlgebra(A); JJ:=J; ll:=1; while Dimension(JJ)>0 do JJ:=ProductSpace(J,JJ); ll:=ll+1; od; return ll; end;

We use this code to calculate the Loewy length of the centre of the group algebra in Chapter 6 as follows:

G:= ; p:= ; #define the group G and prime p K:=GF(p); KG:=GroupRing(K,G); A:=Center(KG); LoewyLength(A);

Note that in the calculations carried out in GAP, we use the Galois ﬁeld of characteristic p as an appropriate splitting ﬁeld. The code above relies on being able to specify the underlying algebra A. When A is the centre of the group algebra, this is straight forward. However, when considering

138 139 the centre of a block, an alternative method has to be used. We present here the GAP code used for the calculations in Chapters 3, 4 and 5.

A.0.1 Multiplications in the centre of the principal block of the group G

The following code is used to calculate the principal block idempotent, and the Loewy structure of Z(kGe0).

Recall that by Remark 1.5.18, for the block B0 = kGe0 of G, the set

D := { Cb(xi) − |C (xi)| eB | xi ∈ P}

is a spanning set for J(Z(kG))e0 = J(Z(kGe0)). In the groups which we are considering, there is only one conjugacy class with p - |C (g)| where p is the characteristic of k; it is referred to as BadConjugacyClass below. Its non-zero size (mod p) has an eﬀect on the multiplication, hence needs to be considered separately.

CALCULATING BLOCK IDEMPOTENT AND STRUCTURE CONSTANTS Arbitrary Group G:= ; p:= ; #Input the group G and prime p t:=CharacterTable(G); Display(t); m:=List(Irr(t),ValuesOfClassFunction); bl:=PrimeBlocks(t,p); l:=SizesConjugacyClasses(t); r:=Size(l); orderG:=Size(t); CharactersInBlock:=Positions(bl.block,1); #characters in principal block s:=l mod p; for i in [2..r] do; if not s[i]=0 then; BadConjugacyClass:=i; fi; od; BadConjugacyClass; #defines the conjugacy class with size NOT divisible by p ======Block idempotent: ======BlockIdempotent:=[]; for j in [1..r] do; a:=0; for i in CharactersInBlock do; a:=a+(m[i][1]*ComplexConjugate(m[i][j])/orderG); od; a:=a mod p; Add(BlockIdempotent,a); od; BlockIdempotent; #gives the coefficients of the block idempotent as a list 140 APPENDIX A. GAP CODE

======Multiplication of 2 class sums in D, including the block idempotent ------for i in [2..r] do; if not i=BadConjugacyClass then; for j in [2..r] do; if not j=BadConjugacyClass then; for n in [1..r] do; c:=0; for x in [1..r] do; a:=BlockIdempotent[x]*ClassStructureCharTable(t,[i,j,x,n])/l[n]; c:=c+a; od; c:=c mod p; if not c=0 then Print(i," ",j," ",n," ",c," "," "); fi; od; fi; od; Print(i, "\n"); fi; od; ======Multiplication of 3 class sums in D, including the block idempotent ------for i in [2..r] do; if not i=BadConjugacyClass then; for j in [2..r] do; if not j=BadConjugacyClass then; for k in [2..r] do; if not k=BadConjugacyClass then; for n in [1..r] do; c:=0; for x in [1..r] do; a:=BlockIdempotent[x]*ClassStructureCharTable(t,[i,j,k,x,n])/l[n]; c:=c+a; od; c:=c mod p; if not c=0 then Print(i," ",j," ",k," ",n," ",c,"\n"); fi; od; fi; od; fi; od; Print(i, "\n"); fi; od; ======Multiplication of 4 class sums in D, including the block idempotent ------for i in [2..r] do; if not i=BadConjugacyClass then; for j in [2..r] do; if not j=BadConjugacyClass then; for k in [2..r] do; if not k=BadConjugacyClass then; for h in [2..r] do; if not h=BadConjugacyClass then; for n in [1..r] do; c:=0; for x in [1..r] do; a:=BlockIdempotent[x]*ClassStructureCharTable(t,[i,j,k,h,x,n])/l[n]; c:=c+a; od; c:=c mod p; if not c=0 then Print(i," ",j," ",k," ",h," ",n," ",c,"\n"); fi; od; fi; od; fi; od; fi; od; Print(i, "\n"); fi; od; ======141

The following code deals with the bad conjugacy class, which has p - |C (g)|. If GAP does not return any outcome, then LL(Z(kGe0)) = 3. for i in [BadConjugacyClass] do; for j in [2..r] do; if not j=BadConjugacyClass then; for k in [2..r] do; if not k=BadConjugacyClass then; for n in [1..r] do; c:=0; for x in [1..r] do; a:=BlockIdempotent[x]*ClassStructureCharTable(t,[i,j,k,x,n])/l[n]; b:=BlockIdempotent[x]*ClassStructureCharTable(t,[1,j,k,x,n])/l[n]; c:=c+a+b; od; c:=c mod p; if not c=0 then Print(i," ",j," ",k," ",n," ",c,"\n"); fi; od; fi; od; fi; od; Print(i, "\n"); od; ------for i in [BadConjugacyClass] do; for j in [BadConjugacyClass] do; for k in [2..r] do; if not k=BadConjugacyClass then; for n in [1..r] do; c:=0; for x in [1..r] do; a:=BlockIdempotent[x]*ClassStructureCharTable(t,[i,j,k,x,n])/l[n]; b:=BlockIdempotent[x]*ClassStructureCharTable(t,[1,1,k,x,n])/l[n]; d:=BlockIdempotent[x]*ClassStructureCharTable(t,[i,1,k,x,n])/l[n]; c:=c+a+b+2*d; od; c:=c mod p; if not c=0 then Print(i," ",j," ",k," ",n," ",c,"\n"); fi; od; fi; od; od; Print(i, "\n"); od; ------i:=BadConjugacyClass; for n in [1..r] do; c:=0; for x in [1..r] do; a1:=BlockIdempotent[x]*ClassStructureCharTable(t,[i,i,i,x,n])/l[n]; a2:=BlockIdempotent[x]*ClassStructureCharTable(t,[i,i,1,x,n])/l[n]; a3:=BlockIdempotent[x]*ClassStructureCharTable(t,[i,1,1,x,n])/l[n]; a4:=BlockIdempotent[x]*ClassStructureCharTable(t,[1,1,1,x,n])/l[n]; c:=c+a1+3*a2+3*a3+a4; od; c:=c mod p; if not c=0 then Print(i," ",n," ",c,"\n"); fi; od;

The following code is used to compute the multiplication C (g)e0 for g ∈ G. ======for i in [2..r] do; if not i=BadConjugacyClass then; 142 APPENDIX A. GAP CODE

for n in [1..r] do; c:=0; for x in [1..r] do; a:=BlockIdempotent[x]*ClassStructureCharTable(t,[i,x,n])/l[n]; c:=c+a; od; c:=c mod p; if not c=0 then Print(i," ",n," ",c," "," "); fi; od; Print(i, "\n"); fi; od; ======

2 A.0.2 Dimension of J (Z(kGe0)) Use ”blockidempotent” and ”BadConjugacyClass” as deﬁned above. We record the multiplication of two elements in D in the matrix A. Its rank over k will then be the 2 dimension of J (Z(kGe0)). A:=[]; for i in [2..r] do; if not i=BadConjugacyClass then; for j in [2..r] do; if not j=BadConjugacyClass then; B:=[]; for n in [1..r] do; c:=0; for x in [1..r] do; a:=BlockIdempotent[x]*ClassStructureCharTable(t,[i,j,x,n])/l[n]; c:=c+a; od; c:=c mod p; Add(B,c); od; B:=[B]; Append(A,B); fi; od; Print(i, "\n"); fi; od; for i in [BadConjugacyClass] do; #adding entries to A for k in [2..r] do; #corresponding to multiplication if not k=BadConjugacyClass then; #with BadConjugacyClass B:=[]; for n in [1..r] do; c:=0; for x in [1..r] do; a:=BlockIdempotent[x]*ClassStructureCharTable(t,[i,k,x,n])/l[n]; b:=BlockIdempotent[x]*ClassStructureCharTable(t,[1,k,x,n])/l[n]; c:=c+a+b; od; c:=c mod p; Add(B,c); od; B:=[B]; Append(A,B); fi; od; Print(i, "\n"); od; for i in [BadConjugacyClass] do; for j in [BadConjugacyClass] do; 143

B:=[]; for n in [1..r] do; c:=0; for x in [1..r] do; a:=BlockIdempotent[x]*ClassStructureCharTable(t,[i,j,x,n])/l[n]; b:=BlockIdempotent[x]*ClassStructureCharTable(t,[1,1,x,n])/l[n]; d:=BlockIdempotent[x]*ClassStructureCharTable(t,[i,1,x,n])/l[n]; c:=c+a+b+2*d; od; c:=c mod p; Add(B,c); od; B:=[B]; Append(A,B); od; Print(i, "\n"); od; ------F:=GF(p); C:=A*One(F);; RankMat(C); # the dimension of J^2(Z(kGe_0)) over F (field of characteristic p) 144 APPENDIX A. GAP CODE

A.0.3 Calculations for normaliser N := NG(P ) The following code is used to calculate the Loewy structure of Z(kN). By remark 1.5.18, the set B := {Cb(xi) − |C (xi)| · 1G | xi ∈ P} forms a basis for J(Z(kN)). As before, in the groups which we are considering, there is only one conjugacy class with p - |C (g)| where p is the characteristic of k; it is referred to as BadConjugacyClass below. Its non-zero size (mod p) has an eﬀect on the multiplication, hence needs to be considered separately.

G:= ; #define the group G and prime p p:= ; S:=SylowSubgroup(G,p); N:=Normaliser(G,S); t:=CharacterTable(N); t:=CharacterTable("GNp"); #Alternatively, use for example t:=CharacterTable("McLN5"); ------Display(t); m:=List(Irr(t),ValuesOfClassFunction); PrimeBlocks(t,p); l:=SizesConjugacyClasses(t); r:=Size(l); s:=l mod p; for i in [2..r] do; if not s[i]=0 then; BadConjugacyClass:=i; fi; od; BadConjugacyClass; #defines the conjugacy class with size NOT divisible by p ------Multiplication of 2 class sums in B ------for i in [2..r] do; if not i=BadConjugacyClass then; for j in [2..r] do; if not j=BadConjugacyClass then; for n in [1..r] do; c:=0; c:=ClassStructureCharTable(t,[i,j,n])/l[n]; c:=c mod p; if not c=0 then Print(i," ",j," ",n," ",c," "," "); fi; od; fi; od; Print(i, "\n"); fi; od; ------Multiplication of 3 class sums in B ------for i in [2..r] do; if not i=BadConjugacyClass then; for j in [2..r] do; if not j=BadConjugacyClass then; for k in [2..r] do; if not k=BadConjugacyClass then; 145

for n in [1..r] do; c:=0; c:=ClassStructureCharTable(t,[i,j,k,n])/l[n]; c:=c mod p; if not c=0 then Print(i," ",j," ",k," ",n," ",c,"\n"); fi; od; fi; od; fi; od; Print(i, "\n"); fi; od; ======

The following code deals with the bad conjugacy class, which has p - |C (g)|. If GAP does not return any outcome, then LL(Z(kN)) = 3.

for i in [BadConjugacyClass] do; for j in [2..r] do; if not j=BadConjugacyClass then; for k in [2..r] do; if not k=BadConjugacyClass then; for n in [1..r] do; c:=0; a:=ClassStructureCharTable(t,[i,j,k,n])/l[n]; b:=ClassStructureCharTable(t,[1,j,k,n])/l[n]; c:=a+b; c:=c mod p; if not c=0 then Print(i," ",j," ",k," ",n," ",c,"\n"); fi; od; fi; od; fi; od; Print(i, "\n"); od; ------for i in [BadConjugacyClass] do; for j in [BadConjugacyClass] do; for k in [2..r] do; if not k=BadConjugacyClass then; for n in [1..r] do; c:=0; a:=ClassStructureCharTable(t,[i,j,k,n])/l[n]; b:=ClassStructureCharTable(t,[1,1,k,n])/l[n]; d:=ClassStructureCharTable(t,[i,1,k,n])/l[n]; c:=a+b+2*d; c:=c mod p; if not c=0 then Print(i," ",j," ",k," ",n," ",c,"\n"); fi; od; fi; od; od; Print(i, "\n"); od; ------i:=BadConjugacyClass; for n in [1..r] do; c:=0; a1:=ClassStructureCharTable(t,[i,i,i,n])/l[n]; a2:=ClassStructureCharTable(t,[i,i,1,n])/l[n]; a3:=ClassStructureCharTable(t,[i,1,1,n])/l[n]; a4:=ClassStructureCharTable(t,[1,1,1,n])/l[n]; c:= a1+3*a2+3*a3+a4; c:=c mod p; if not c=0 then Print(i," ",n," ",c,"\n"); fi; od; 146 APPENDIX A. GAP CODE

A.0.4 Dimension of J 2(Z(kN)) Use ”BadConjugacyClass” as deﬁned above. We record the multiplication of two elements in B in the matrix A. Its rank over k will then be the dimension of J 2(Z(kN)).

D:=[]; for i in [1..r] do; Add(D,0); od; A:=[]; for n in [2..r] do; if not n=BadConjugacyClass then; B:=MatClassMultCoeffsCharTable(t,n) mod p; B[BadConjugacyClass]:=D; B[1]:=D; Append(A,B); fi; od; for i in [BadConjugacyClass] do; # adding entries to A for k in [2..r] do; # corresponding to multiplication if not k=BadConjugacyClass then; # with BadConjuagcyClass B:=[]; for n in [1..r] do; c:=0; a:=ClassStructureCharTable(t,[i,k,n])/l[n]; b:=ClassStructureCharTable(t,[1,k,n])/l[n]; c:=a+b; c:=c mod p; Add(B,c); od; B:=[B]; Append(A,B); fi; od; Print(i, "\n"); od; for i in [BadConjugacyClass] do; for k in [BadConjugacyClass] do; B:=[]; for n in [1..r] do; c:=0; a:=ClassStructureCharTable(t,[i,k,n])/l[n]; b:=ClassStructureCharTable(t,[1,k,n])/l[n]; d:=ClassStructureCharTable(t,[1,1,n])/l[n]; c:=a+2*b+d; c:=c mod p; Add(B,c); od; B:=[B]; Append(A,B); od; Print(i, "\n"); od; ------F:=GF(p); C:=A*One(F);; RankMat(C); #the dimension of J^2(Z(kN)) over F (field of characteristic p) Appendix B

Some structure constants in Ree groups

In this chapter we give the structure constants calculated in Section 4.3.3. Here G = 2 k 2 G2(q) and m = 3 , where q = 3m . Therefore most structure constants given in the tables below are zero modulo 3. a(X, X, z) mod 3 a(X,X, 1) (q3 + 1) ∗ (q − 1) 2 a(X,X,Ra) 2 ∗ q − 2 1 a(X,X,Sa) 12 ∗ m2 + 4 1 2 a(X,X,Vi) 3 ∗ m − 3 ∗ m + 1 1 2 a(X,X,Wi) 3 ∗ m + 3 ∗ m + 1 1 a(X,X,JT ) 0 0 a(X,X,JT −1) 0 0 a(X,X,JRa 0 0 a(X,X,JSa) 0 0 a(X,X,J) 0 0 a(X, T, z) mod 3 a(X,T,Ra) 9/2 ∗ m4 − 3 ∗ m2 + 1/2 2 a(X,T,Sa) 9/2 ∗ m4 − 3 ∗ m2 − 3/2 0 4 2 a(X,T,Vi) 9/2 ∗ m − 3 ∗ m + 3/2 0 4 2 a(X,T,Wi) 9/2 ∗ m − 3 ∗ m − 3/2 0 a(X,T,JT ) 9 ∗ m4 + 3 ∗ m2 0 a(X,T,JT −1 0 0 a(X,T,JRa) 9/2 ∗ m4 − 1/2 1 a(X,T,JSa) 9/2 ∗ m4 + 3 ∗ m2 + 1/2 2 a(X,T,J) 0 0 a(X,T,T ) 3 ∗ m2 − 1 2 a(X,T,T −1) 0 0

147 148 APPENDIX B. SOME STRUCTURE CONSTANTS IN REE GROUPS

a(T, Y T, z) a(T,YT,Ra) 27/2 ∗ m8 − 9/2 ∗ m6 + 3/2 ∗ m4 − 1/2 ∗ m2 a(T,YT,Sa) 27/2 ∗ m8 − 9/2 ∗ m6 − 9 ∗ m5 + 3/2 ∗ m4 − 3 ∗ m3 + 3/2 ∗ m2 8 6 5 4 3 a(T,YT,Vi) 27/2 ∗ m − 9/2 ∗ m + 9/2 ∗ m − 3 ∗ m + 3/2 ∗ m 8 6 5 4 3 a(T,YT,Wi) 27/2 ∗ m − 9/2 ∗ m + 9/2 ∗ m + 6 ∗ m + 3/2 ∗ m a(T,YT,JT ) 27/2 ∗ m8 − 9 ∗ m6 − 9/2 ∗ m4 a(T,YT,JT −1) 27/2 ∗ m8 − 9 ∗ m5 − 3/2 ∗ m4 − 3 ∗ m3 a(T,YT,JRa) 27/2 ∗ m8 − 9/2 ∗ m6 − 3/2 ∗ m4 + 1/2 ∗ m2 a(T,YT,JSa) 27/2 ∗ m8 − 9/2 ∗ m6 − 9 ∗ m5 − 9/2 ∗ m4 − 3 ∗ m3 − 1/2 ∗ m2 a(T,YT,J) 27/2 ∗ m8 + 27/2 ∗ m7 − 3/2 ∗ m4 − 3/2 ∗ m3

a(T,YT −1, z) a(T,YT −1,Ra) 27/2 ∗ m8 − 9/2 ∗ m6 + 3/2 ∗ m4 − 1/2 ∗ m2 a(T,YT −1,Sa) 27/2 ∗ m8 − 9/2 ∗ m6 + 9 ∗ m5 + 3/2 ∗ m4 + 3 ∗ m3 + 3/2 ∗ m2 −1 8 6 5 4 3 a(T,YT ,Vi) 27/2 ∗ m − 9/2 ∗ m − 9/2 ∗ m + 6 ∗ m − 3/2 ∗ m −1 8 6 5 4 3 a(T,YT ,Wi) 27/2 ∗ m − 9/2 ∗ m − 9/2 ∗ m − 3 ∗ m − 3/2 ∗ m a(T,YT −1,JT ) 27/2 ∗ m8 − 9 ∗ m6 − 9/2 ∗ m4 a(T,YT −1,JT −1) 27/2 ∗ m8 + 9 ∗ m5 − 3/2 ∗ m4 + 3 ∗ m3 a(T,YT −1,JRa) 27/2 ∗ m8 − 9/2 ∗ m6 − 3/2 ∗ m4 + 1/2 ∗ m2 a(T,YT −1,JSa) 27/2 ∗ m8 − 9/2 ∗ m6 + 9 ∗ m5 − 9/2 ∗ m4 + 3 ∗ m3 − 1/2 ∗ m2 a(T,YT −1,J) 27/2 ∗ m8 − 27/2 ∗ m7 − 3/2 ∗ m4 + 3/2 ∗ m3

a(T, JT, z) a(T,JT,Ra) 81/4 ∗ m8 − 27/4 ∗ m6 − 9/4 ∗ m4 + 3/4 ∗ m2 a(T,JT,Sa) 81/4 ∗ m8 + 27/4 ∗ m6 − 9/4 ∗ m4 − 3/4 ∗ m2 8 6 5 4 2 a(T,JT,Vi) 81/4 ∗ m − 27/2 ∗ m + 27/4 ∗ m − 9/4 ∗ m + 3/2 ∗ m − 3/4 ∗ m 8 6 5 4 2 a(T,JT,Wi) 81/4 ∗ m − 27/2 ∗ m − 27/4 ∗ m − 9/4 ∗ m + 3/2 ∗ m + 3/4 ∗ m a(T,JT,JT ) 81/4 ∗ m8 − 27/2 ∗ m6 − 9/2 ∗ m4 − 9/4 ∗ m2 a(T,JT,JT −1) 81/4 ∗ m8 + 27/2 ∗ m6 + 27/2 ∗ m4 + 3/4 ∗ m2 a(T,JT,JRa) 81/4 ∗ m8 − 27/4 ∗ m6 + 9/4 ∗ m4 − 3/4 ∗ m2 a(T,JT,JSa) 81/4 ∗ m8 + 27/4 ∗ m6 + 27/4 ∗ m4 + 9/4 ∗ m2 a(T,JT,J) 0 149 a(T,JT −1, z) a(T,JT −1,Ra) 81/4 ∗ m8 − 27/4 ∗ m6 − 9/4 ∗ m4 + 3/4 ∗ m2 a(T,JT −1,Sa) 81/4 ∗ m8 − 81/4 ∗ m6 − 81/4 ∗ m4 − 15/4 ∗ m2 −1 8 5 4 3 2 a(T,JT ,Vi) 81/4 ∗ m − 27/4 ∗ m + 27/4 ∗ m − 9/2 ∗ m + 3 ∗ m − 3/4 ∗ m −1 8 5 4 3 2 a(T,JT ,Wi) 81/4 ∗ m + 27/4 ∗ m + 27/4 ∗ m + 9/2 ∗ m + 3 ∗ m + 3/4 ∗ m a(T,JT −1,JT ) 81/4 ∗ m8 − 27/2 ∗ m6 − 9/2 ∗ m4 − 9/4 ∗ m2 a(T,JT −1,JT −1) 81/4 ∗ m8 − 27/2 ∗ m6 − 9/2 ∗ m4 − 9/4 ∗ m2 a(T,JT −1,JRa) 81/4 ∗ m8 − 27/4 ∗ m6 + 9/4 ∗ m4 − 3/4 ∗ m2 a(T,JT −1,JSa) 81/4 ∗ m8 − 81/4 ∗ m6 − 45/4 ∗ m4 − 3/4 ∗ m2 a(T,JT −1,J) 81/2 ∗ m8 + 27/2 ∗ m6 − 9/2 ∗ m4 − 3/2 ∗ m2

a(Y, T, z) a(Y,T,Ra) 27/2 ∗ m8 − 9/2 ∗ m6 + 3/2 ∗ m4 − 1/2 ∗ m2 a(Y,T,Sa) 27/2 ∗ m8 − 9/2 ∗ m6 + 21/2 ∗ m4 + 9/2 ∗ m2 8 6 4 3 2 a(Y,T,Vi) 27/2 ∗ m − 9/2 ∗ m − 3 ∗ m + 9/2 ∗ m − 3/2 ∗ m 8 6 4 3 2 a(Y,T,Wi) 27/2 ∗ m − 9/2 ∗ m − 3 ∗ m − 9/2 ∗ m − 3/2 ∗ m a(Y,T,JT ) 27/2 ∗ m8 + 9/2 ∗ m6 + 9/2 ∗ m4 + 3/2 ∗ m2 a(Y,T,JT −1 27/2 ∗ m8 − 27/2 ∗ m6 − 3/2 ∗ m4 + 3/2 ∗ m2 a(Y,T,JRa) 27/2 ∗ m8 − 9/2 ∗ m6 − 3/2 ∗ m4 + 1/2 ∗ m2 a(Y,T,JSa) 27/2 ∗ m8 − 9/2 ∗ m6 + 9/2 ∗ m4 + 5/2 ∗ m2 a(Y,T,J) 27/2 ∗ m8 − 27/2 ∗ m6 − 3/2 ∗ m4 + 3/2 ∗ m2 a(Y,T,T ) 27/2 ∗ m8 + 9/2 ∗ m4

a(Y, X, z) a(Y,X,Ra) 9 ∗ m6 − 6 ∗ m4 + m2 a(Y,X,Sa) 9 ∗ m6 − 24 ∗ m4 − 9 ∗ m2 6 4 3 2 a(Y,X,Vi) 9 ∗ m + 3 ∗ m − 6 ∗ m + 3 ∗ m 6 4 3 2 a(Y,X,Wi) 9 ∗ m + 3 ∗ m + 6 ∗ m + 3 ∗ m a(Y,X,JT ) 9 ∗ m6 − 6 ∗ m4 − 3 ∗ m2 a(Y,X,JT −1 9 ∗ m6 − 6 ∗ m4 − 3 ∗ m2 a(Y,X,JRa) 9 ∗ m6 − m2 a(Y,X,JSa) 9 ∗ m6 − 12 ∗ m4 − 5 ∗ m2 a(Y,X,J) 27 ∗ m6 − 3 ∗ m2 a(Y,X,T ) 9 ∗ m6 − 9 ∗ m4 a(Y,X,Y ) 9 ∗ m6 + 3 ∗ m2 − 1 150 APPENDIX B. SOME STRUCTURE CONSTANTS IN REE GROUPS

a(T, T, z) mod 3 a(T,T, 1) 0 0 a(T,T,Ra) 27/4 ∗ m6 − 9/4 ∗ m4 + 3/4 ∗ m2 − 1/4 2 a(T,T,Sa) 27/4 ∗ m6 + 9/4 ∗ m4 + 9/4 ∗ m2 + 3/4 0 6 5 4 3 2 a(T,T,Vi) 27/4 ∗ m + 27/4 ∗ m − 9/2 ∗ m − 9/4 ∗ m + 9/4 ∗ m 0 6 5 4 3 2 a(T,T,Wi) 27/4 ∗ m − 27/4 ∗ m − 9/2 ∗ m + 9/4 ∗ m + 9/4 ∗ m 0 a(T,T,JT ) 27/4 ∗ m6 + 9/2 ∗ m4 + 3/4 ∗ m2 0 a(T,T,JT −1 27/4 ∗ m6 − 9/2 ∗ m4 − 9/4 ∗ m2 0 a(T,T,JRa) 27/4 ∗ m6 − 9/4 ∗ m4 − 3/4 ∗ m2 + 1/4 1 a(T,T,JSa) 27/4 ∗ m6 + 9/4 ∗ m4 − 3/4 ∗ m2 − 1/4 2 a(T,T,J) 0

a(T,T −1, z) mod 3 a(T,T −1,Ra) 27/4 ∗ m6 − 9/4 ∗ m4 + 3/4 ∗ m2 − 1/4 2 a(T,T −1,Sa) 27/4 ∗ m6 − 27/4 ∗ m4 − 3/4 ∗ m2 + 3/4 0 −1 6 5 3 2 a(T,T ,Vi) 27/4 ∗ m − 27/4 ∗ m − 9/4 ∗ m + 15/4 ∗ m − 3/2 ∗ m 0 −1 6 5 3 2 a(T,T ,Wi) 27/4 ∗ m + 27/4 ∗ m + 9/4 ∗ m + 15/4 ∗ m + 3/2 ∗ m 0 a(T,T −1,JT ) 27/4 ∗ m6 − 9/2 ∗ m4 − 9/4 ∗ m2 0 a(T,T −1,JT −1) 27/4 ∗ m6 − 9/2 ∗ m4 − 9/4 ∗ m2 0 a(T,T −1,JRa) 27/4 ∗ m6 − 9/4 ∗ m4 − 3/4 ∗ m2 + 1/4 1 a(T,T −1,JSa) 27/4 ∗ m6 − 27/4 ∗ m4 − 15/4 ∗ m2 − 1/4 2 a(T,T −1,J) 27/2 ∗ m6 − 3/2 ∗ m2 0 a(T,T −1,T −1) 9/4 ∗ m4 − 9/4 ∗ m2 0 a(T,T −1,Y ) 27/4 ∗ m6 + 9/4 ∗ m2 0