International Electronic Journal of Algebra Volume 18 (2015) 21-33
UNITS IN Fqk (Cp or Cq)
N. Makhijani, R. K. Sharma and J. B. Srivastava
Received: 5 February 2014 Communicated by A. C¸i˘gdem Ozcan¨
k Abstract. Let q be a prime, Fqk be a finite field having q elements and n q −1 r Cn or Cq be a group with presentation h a, b | a , b , b ab = a i, where (n, rq) = 1 and q is the multiplicative order of r modulo n. In this paper, we address the problem of computing the Wedderburn decomposition of the
group algebra Fqk (Cn or Cq) modulo its Jacobson radical. As a consequence,
the structure of the unit group of Fqk (Cp or Cq) is obtained when p is a prime different from q.
Mathematics Subject Classification (2010): 16U60, 16S34, 20C05 Keywords: Unit group, group algebra, Wedderburn decomposition
1. Introduction
Let FG be the group algebra of a finite group G over a field F and U(FG) be its unit group. The study of the group of units is one of the classical topics in group ring theory and has applications in coding theory (cf. [17, 28]) and cryptography (cf. [18]). Results obtained in this direction are useful for the investigation of Lie properties of group rings, isomorphism problem and other open questions in this area [6]. In [3], Bovdi gave a comprehensive survey of results concerning the group of units of a modular group algebra of characteristic p. There is a long tradition on the study of the unit group of finite group algebras [4, 5, 12–16, 22, 25, 26]. In general, the structure of U(FG) is elusive if |G| = 0 in F . Let J(FG) be the Jacobson radical of FG. Then
ψ FG 1 −→ 1 + J(FG) −→inc U(FG) −→ U −→ 1 J(FG) where ψ(x) = x + J(FG), ∀x ∈ U(FG), is a short exact sequence of groups and if F is a perfect field, then by Wedderburn-Malcev Theorem [9, Theorem 72.19], there exists a semisimple subalgebra B of FG such that FG = B ⊕ J(FG) showing that the above sequence is split and hence
U(FG) =∼ (1 + J(FG)) o U (F G/J(FG)) 22 N. MAKHIJANI, R. K. SHARMA AND J. B. SRIVASTAVA
Thus a good description of the Wedderburn decomposition of F G/J(FG) is useful for studying the unit group of FG. The computation of the Wedderburn decomposition of finite semisimple group algebras and in particular, of the primitive central idempotents, has attracted the attention of several authors (cf. [1, 2, 7, 27]). This raises the following question: Is it possible to efficiently compute the decomposition of F G/J(FG) when FG is a finite group algebra that is not semi-simple? An expression for the decomposition of
F2D2p/J(F2D2p) was obtained by Kaur and Khan in [21], where D2p is a dihedral group of order 2p and p is an odd prime. Recently, the authors generalized this expression and computed the decomposition of F2k D2n/J(F2k D2n) for an arbitrary odd number n in [24]. In this paper, we focus on the computation of the Wedderburn decomposition of the group algebra Fqk (Cnor Cq) modulo its Jacobson radical when n q −1 r q is a prime and Cn or Cq is the group with presentation h a, b | a , b , b ab = a i such that (n, rq) = 1 and the multiplicative order of r modulo n is q. It was proved by Kaur and Khan during the conference on Groups, Group Rings and Related Topics (cf. [20]) that if p and q are distinct primes and F is a finite field of characteristic q having a primitive pth root of unity, then
U(F (Cp o Cq)) ∗ p−1 =∼ F × GL(q, F ) q 1 + J(F (Cp o Cq)) The structure of unit group is however open to be explored in the absence of a primitive pth root of unity in F . We use the decomposition obtained in Section 4 to determine the structure of the unit group of F (Cp o Cq) for any finite field F of characteristic q.
2. Notations
We introduce some basic notations where l and m are coprime integers, R is a ring, g ∈ G and X is any subset of G.
Zm ring of integers modulo m
ordm(l) multiplicative order of l modulo m
irrF (α) minimal polynomial of α over F ϕ(n) Euler’s totient function of n F ∗ F \{0} Xb P x x∈X gb hdg i UNITS IN Fqk (Cp or Cq ) 23
o(g) order of g [ g ] conjugacy class of g Gm external direct product of m copies of G Rm external direct sum of m copies of R M(m, F ) algebra of all m × m matrices over F GL(m, F ) general linear group of all m × m invertible matrices over F
3. Preliminaries
The following results will be useful for our investigation.
Proposition 3.1. [19, Chapter 1, Proposition 6.16] Let f : R1 → R2 be a sur- jective homomorphism of rings. Then f (J(R1)) ⊆ J(R2) with equality if ker f ⊆
J(R1).
Remark 3.2. Note that if we add the semi-simplicity of the ring R2 in the above proposition, then J(R1) ⊆ ker f.
Theorem 3.3. Let E = F (ξ)/F be a finite separable extension, K be any field extension of F and g(X) = irrF (ξ). If r Y g(X) = gi(X) i=1 as a product of irreducible polynomials gi(X) ∈ K[X], then ∼ r K ⊗F E = ⊕ K(ξi) i=1 as K-algebras, where ξi is a root of gi(X) in an algebraic closure L of K.
Proof. For each i, 1 ≤ i ≤ r, the map λi : K ×E → K(ξi) given by the assignment
(α, f(ξ)) 7→ αf(ξi) ∀ α ∈ K, f(X) ∈ F [X]
∗ is F -bilinear and hence induces a K-algebra homomorphism λi : K ⊗F E → K(ξi) such that
α ⊗ f(ξ) 7→ αf(ξi) ∗ ∗ Evidently λi is onto. Therefore ker λi is a maximal ideal of K ⊗F E for all i, 1 ≤ ∗ ∗ ∗ ∗ i ≤ r and ker λi + ker λj = K ⊗F E for any i 6= j as 1 ⊗ gi(ξ) ∈ ker λi \ ker λj . It follows from [8, Chapter 5, Theorem 2.2] that the K-algebra homomorphism r λ : K ⊗F E → ⊕ K(ξi) defined by i=1 ∗ ∗ λ(A) = (λ1(A), . . . , λr(A)) ∀ A ∈ K ⊗F E 24 N. MAKHIJANI, R. K. SHARMA AND J. B. SRIVASTAVA
r P r is onto. Since dimK K⊗F E = dimF E = deg g(X) = deg gi(X) = dimK ⊕ K(ξi), i=1 i=1 the proof follows.
Theorem 3.4. [23, Theorem 2.21] The distinct automorphisms of Fuk over Fu uj are exactly the mappings σ0, . . . , σk−1, defined by σj(α) = α for α ∈ Fuk and 0 ≤ j ≤ k − 1.
Thus if v = uk, then (m,k) ⊗ m =∼ k (3.1) Fv Fu Fu Fvom k as Fv-algebras where om = ordum−1(v) = m/(m, k).
Theorem 3.5. [10, Theorem 7.9] Let A be an F -algebra, E be an extension field of F and suppose that for each simple A-module M, the E ⊗F A module E ⊗F M is semi-simple (This hypothesis certainly holds whenever E is a finite separable extension of F ). Then
(1) J(E ⊗F A) = E ⊗F J(A) A ∼ E ⊗F A (2) E ⊗F = J(A) E ⊗F J(A)
As a consequence, if A is a finite dimensional F -algebra, then dimE J(E⊗F A) = dimF J(A).
The following results are due to Ferraz.
Let G be a finite group and char F = p. Also let s be the L.C.M. of the orders of th the p-regular elements of G, ξ be a primitive s root of unity over F and TG,F be the multiplicative group consisting of those integers t, taken modulo s, for which ξ 7→ ξt defines an automorphism of F (ξ) over F .
If F = Fu, then by Theorem 3.4,
c−1 TG,F = {1, u, ··· , u } mod s where c = ords(u).
We denote the sum of all conjugates of g ∈ G by γg.
Definition 3.6. If g ∈ G is a p-regular element, then the cyclotomic F -class of γg is defined to be the set
SF (γg) = { γgt | t ∈ TG,F }
Proposition 3.7. [11, Proposition 1.2] The number of simple components of F G/J(FG) is equal to the number of cyclotomic F -classes in G. UNITS IN Fqk (Cp or Cq ) 25
Theorem 3.8. [11, Theorem 1.3] Suppose that Gal(F (ξ): F ) is cyclic. Let w be the number of cyclotomic F -classes in G. If K1, ··· ,Kw are the simple components of Z(F G/J(FG)) and S1, ··· ,Sw are the cyclotomic F -classes of G, then with a suitable re-ordering of indices, | Si | = [Ki : F ].
Note that the conjugacy class of a p-regular element of G is referred to as a p-regular conjugacy class.
4. Wedderburn decomposition
In this section, we compute a general expression for the Wedderburn decompo- sition of the group algebra Fqk (Cn or Cq) modulo its Jacobson radical.
Lemma 4.1. Let F be a finite field of characteristic q containing a primitive nth root of unity ζ and G = Cn or Cq. Then
1+P ϕ(d) P ϕ(d) F G/J(FG) =∼ F d∈An ⊕ M(q, F ) d∈Bn q where
An = { d | d > 1, d | n and ordd(r) = 1 } ,
Bn = { d | d > 1, d | n and ordd(r) = q } .
Proof. A q-regular conjugacy class of G is either {1} or of the form
i i a if o(a ) ∈ An [ ai ] = n i ri rq−1i o i a , a , ··· , a if o(a ) ∈ Bn showing that if d ∈ Bn, then there are sd = ϕ(d)/q distinct conjugacy classes in Im G containing elements of order d, each of size q. Let a d 1 ≤ m ≤ sd be the representatives of these classes and Cn = { (d, m) | d ∈ Bn, 1 ≤ m ≤ sd }.
Define the following F -algebra homomorphisms:
(a) φ1 : FG → F by the assignment a 7→ 1, b 7→ 1.
m (b) For each (d, m) ∈ Cn, θd : FG → M(q, F ) by
Im ζ d 0 0 ··· 0 0 0 ··· 0 1 Im r 0 ζ d 0 ··· 0 1 0 ··· 0 0 a 7→ . . . . , b 7→ . . . . . . . .. 0 . . .. . 0 Im rq−1 0 0 0 ··· ζ d 0 0 0 1 0 q×q q×q 26 N. MAKHIJANI, R. K. SHARMA AND J. B. SRIVASTAVA
Firstly suppose that An 6= ∅ and for each d ∈ An, define the F -algebra homomor- phism ϕ(d) φd : FG → F by the assignment n d i a 7→ ζ i ∈ U( ) , b 7→ ( 1, ··· , 1 ) Zd | {z } ϕ(d) P ϕ(d) P s We claim that if θ : FG → F ⊕ F d∈An ⊕ M(q, F ) d∈Bn d is defined as
m θ = φ1 ⊕ ⊕ φd ⊕ ⊕ θd , d∈An (d,m)∈Cn
! X then dimF ker θ = (q − 1) 1 + ϕ(d) .
d∈An
q−1 n−1 n−1 X X j j i X j i Let X = αi b a ∈ ker θ and Fj(X) = αi X ∈ F [X] for all j, 0 ≤ j ≤ j=0 i=0 i=0 q − 1.
Then
q−1 n−1 X X j (a) αi = 0 j=0 i=0
q−1 X m j m (b) θd (b) Fj (θd (a)) = O ∀ (d, m) ∈ Cn j=0 Imri ⇒ Fj(ζ d ) = 0 ∀ (d, m) ∈ Cn, 0 ≤ i, j ≤ q − 1
sd q−1 Q Q Imri Since Φd(X) = (X − ζ d ) ∀ d ∈ Bn, therefore m=1 i=0 ! Y Fj(X) = Gj(X) Φd(X)
d∈Bn
m0 X j i 0 X X for some Gj(X) = βi X ∈ F [X], where m = n−1− ϕ(d) = ϕ(d). i=0 d∈Bn d∈An
q−1 X n i (c) Fj ζ d = 0 ∀ i ∈ U(Zd), d ∈ An. j=0 q−1 ! X Y ⇒ Fj(X) = Φd(X) g(X), for some g(X) ∈ F [X].
j=0 d∈An UNITS IN Fqk (Cp or Cq ) 27
q−1 ! ! X Y Y ⇒ Gj(X) Φd(X) = Φd(X) g(X)
j=0 d∈Bn d∈An
Due to degree constraints, we have ! Q g(X) = α Φd(X) for some α ∈ F . d∈Bn
Using (a), g(1) = 0 and hence
⇒ g(X) = 0 q−1 X ⇒ Gj(X) = 0 j=0 q−1 X j 0 ⇒ βi = 0 ∀ 0 ≤ i ≤ m j=0 ! 0 X ⇒ dim F ker θ = (q − 1)(1 + m ) = (q − 1) 1 + ϕ(d)
d∈An Hence the claim follows.
P ϕ(d) Evidently if An = ∅ and θ : FG → F ⊕ M(q, F ) d∈Bn q is defined as m θ = φ1 ⊕ ⊕ θd (d,m)∈Cn then dim F ker θ = q − 1.
Thus in either case, θ is onto and hence J(FG) ⊆ ker θ showing that θ∗ : F G/J(FG) → θ(FG) defined by
θ∗ (X + J(FG)) = θ(X) ∀ X ∈ FG is a well-defined surjective F -algebra homomorphism.
As F G/J(FG) is semi-simple, it follows that
F G/J(FG) =∼ C ⊕ θ(FG) for the semi-simple F -algebra C = ker θ∗.
But TG,F = {1} mod n. Thus the cyclotomic F -classes in G are precisely the q-regular conjugacy classes in G and by Proposition 3.7, the number of simple components in the decomposition of F G/J(FG) is X X ϕ(d) 1 + ϕ(d) + q d∈An d∈Bn 28 N. MAKHIJANI, R. K. SHARMA AND J. B. SRIVASTAVA which is same as the number of simple components in θ (FG). Hence
F G/J(FG) =∼ θ(FG) and the proof follows. The decomposition may vary if F does not contain a primitive nth root of unity.
Theorem 4.2. If u = qk, then xd yd ∼ o FuG/J(FuG) = Fu ⊕ ⊕ ( Fu d ) ⊕ ⊕ M (q, Fuid ) (4.1) d∈An d∈Bn where
(a) od = ordd(u) ∀ d | n, d > 1.
ϕ(d) (b) xd = ∀ d ∈ An ( if An 6= ∅). od
(c) for each d ∈ Bn, id is the least positive integer such that
uid ≡ rj mod d
ϕ(d) for some 0 ≤ j ≤ q − 1 and yd = . q id
Proof. Let g ∈ G \{1} be a q-regular element and d = o(g). The following hold:
(1) If An 6= ∅ and d ∈ An, then [˙ n o j SFu (γg) = γgu 0 ≤ j ≤ od−1
(2) However if d ∈ Bn, then Im [ g ] = [ a d ] for some m, 1 ≤ m ≤ sd For any l, l0 ∈ N,
γ l = γ l0 gu gu 0 ulIm ul Im ⇔ [ a d ] = [ a d ]
0 ⇔ u|l−l | ≡ rj mod d for some j, 0 ≤ j ≤ q − 1
Therefore [˙ n o j SFu (γg) = γgu 0 ≤ j ≤ id−1 Thus there are
(a) xd distinct cyclotomic Fu-classes in G, each of order od, for all d ∈ An (if
An 6= ∅).
(b) yd cyclotomic Fu-classes, each of order id, for all d ∈ Bn. UNITS IN Fqk (Cp or Cq ) 29
In view of the Wedderburn structure theorem and Theorem 3.8,
FuG/J(FuG) (4.2)
xd yd ∼ o = Fu ⊕ ⊕ ⊕ M (nl, Fu d ) ⊕ ⊕ ⊕ M (mj, Fuid ) d∈An l=1 d∈Bn j=1 for some nl, mj ≥ 1. By [10, Theorem 7.9], it follows that
o o ∼ o Fu n G/J(Fu n G) = Fu n ⊗Fu FuG/J(FuG)
xd ∼ o o o = Fu n ⊕ ⊕ ⊕ M (nl, Fu n ⊗Fu Fu d ) d∈An l=1
yd o i ⊕ ⊕ ⊕ M (mj, Fu n ⊗Fu Fu d ) d∈Bn j=1
Since Fuon contains a primitive nth root of unity, therefore the decomposition of
Fuon G/J(Fuon G) can be obtained using the previous lemma. As a consequence of equation (3.1) and the uniqueness of Wedderburn decomposition, we have nl, mj = 1 or q.
Let κ : FuG → Fu be the Fu-algebra homomorphism, coming from the trivial representation of G and K = ker κ.
If An 6= ∅ and d ∈ An, then
xd Y i Φd(X) = fd(X) i=1 i where fd(X) ∈ Fu[X] is an irreducible polynomial of degree od for each 1 ≤ i ≤ xd. i Now for each (d, i) ∈ Gn = { (d, i) | d ∈ An, 1 ≤ i ≤ xd }, let ξd ∈ Fuod be a root i i of fd(X) and κd : FuG → Fuod be an Fu-algebra homomorphism obtained from the assignment i a 7→ ξd, b 7→ 1 i i i Observe that for all (d, i) ∈ Gn, κd is onto and hence Kd = ker κd is a maximal ideal of FuG. Since
i i i0 0 0 0 0 fd(a) ∈ Kd \ Kd0 ∀ (d, i), (d , i ) ∈ Gn and (d, i) 6= (d , i ), it follows that i K,Kd (d, i) ∈ Gn is a collection of pairwise co-maximal ideals of FuG. Therefore by Chinese remain- der theorem [8, Chapter 5, Theorem 2.2], it follows that G Fu ∼ xd ! = Fu ⊕ ⊕ ( Fuod ) d∈An T T i K Kd (d,i)∈Gn 30 N. MAKHIJANI, R. K. SHARMA AND J. B. SRIVASTAVA
Using this with Remark 3.2, there exists a surjective Fu-algebra homomorphism FuG xd λ : M = Fu ⊕ ⊕ ( Fuod ) J(FuG) d∈An showing that M occurs in the decomposition of FuG/J(FuG) given in equation (4.2).
Recall that nl, mj = 1 or q and ! X 2 X 1 + od × xd + q id × yd
d∈An d∈Bn ! X X = 1 + ϕ(d) + q ϕ(d)
d∈An d∈Bn ! X = n + (q − 1) ϕ(d)
d∈Bn ! X = nq − (q − 1) n − ϕ(d)
d∈Bn
o = |G| − dim Fuon J(Fu n G)
= |G| − dim Fu J(FuG) using [10, Theorem 7.9] Therefore xd yd ∼ o FuG/J(FuG) = Fu ⊕ ⊕ ( Fu d ) ⊕ ⊕ M (q, Fuid ) d∈An d∈Bn
Via similar arguments the theorem holds true even when An = ∅.
5. Main result
The main result of the paper is as follows.
k Theorem 5.1. Let u = q and G = Cn or Cq where (n, q) = 1. If ordd(r) = q for every divisor d(> 1) of n, then
∼ k(q−1) Y yd U (FuG) = Cq × Cu−1 × GL(q, Fuid ) d >1 d | n where id is the least positive integer such that
uid ≡ rj mod d
ϕ(d) for some 0 ≤ j ≤ q − 1 and yd = . In particular, q id
∼ k(q−1) yp U (Fu(Cp o Cq)) = Cq o ( Cu−1 × GL (q, Fuip ) ) UNITS IN Fqk (Cp or Cq ) 31 where p is a prime different from q.
q−1 i X Proof. Since b ba ∈ Z(FuG) ∀ 0 ≤ i ≤ q − 1, therefore if αi = 0, then for any i=0 X ∈ FuG, q−1 !q X i 1 + X αi b ba = 1 i=0 showing that ( q−1 q−1 )
X i X αi b ba αi ∈ Fu such that αi = 0 ⊆ J(FuG). i=0 i=0
But from the proof of Theorem 4.2, it follows that dim Fu J(FuG) = q − 1 and hence equality. ∼ k(q−1) Thus 1 + J(FuG) = Cq . Since J(FuG) ⊆ Z(FuG) and ∼ U(FuG) = (1 + J(FuG)) o U (FuG/J(FuG)) therefore the proof follows by using Theorem 4.2.
k Corollary 5.2. Let D2n be the dihedral group of order 2n, n odd and u = 2 . Then
∼ k Y yd U (FuD2n) = C2 × Cu−1 × GL(2, Fuid ) d >1 d | n where od = ordd(u), od if o is even and uod/2 ≡ −1 mod d 2 d id = od otherwise
ϕ(d) and yd = . 2id
Remark 5.3. This is in accordance with the structure of the unit group of F2k D2n determined by the authors in [24] for odd n.
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N. Makhijani, R. K. Sharma and J. B. Srivastava Department of Mathematics Indian Institute of Technology New Delhi, India e-mails: [email protected] (N. Makhijani) [email protected] (R. K. Sharma) [email protected] (J. B. Srivastava)