N. Makhijani, R. K. Sharma and J. B. Srivastava

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1 International Electronic Journal of Algebra Volume UNITS IN F q kc p r C q N. Makhijani, R. K. Sharma and J. B. Srivastava Received: 5 February 2014 Communicated by A. Çiğdem Özcan Abstract. Let q be a prime, F q k be a finite field having q k elements and C n r C q be a group with presentation a, b a n, b q, b 1 ab = a r, 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 F q k C n r C q modulo its Jacobson radical. As a consequence, the structure of the unit group of F q k C p r C q 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 F G be the group algebra of a finite group G over a field F and UF G 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 UF G is elusive if G = 0 in F. Let JF G be the Jacobson radical of F G. Then JF G inc UF G ψ U F G JF G 1 where ψx = x + JF G, x UF G, 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 F G such that F G = B JF G showing that the above sequence is split and hence UF G = 1 + JF G U F G/JF G

2 22 N. MAKHIJANI, R. K. SHARMA AND J. B. SRIVASTAVA Thus a good description of the Wedderburn decomposition of F G/JF G is useful for studying the unit group of F G. 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/JF G when F G is a finite group algebra that is not semi-simple? An expression for the decomposition of F 2 D 2p /JF 2 D 2p was obtained by Kaur and Khan in [21], where D 2p is a dihedral group of order 2p and p is an odd prime. Recently, the authors generalized this expression and computed the decomposition of F 2 kd 2n /JF 2 kd 2n for an arbitrary odd number n in [24]. In this paper, we focus on the computation of the Wedderburn decomposition of the group algebra F q kc n r C q modulo its Jacobson radical when q is a prime and C n r C q is the group with presentation a, b a n, b q, b 1 ab = a r 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 p th root of unity, then UF C p C q 1 + JF C p C q = F GLq, F p 1 q The structure of unit group is however open to be explored in the absence of a primitive p th root of unity in F. We use the decomposition obtained in Section 4 to determine the structure of the unit group of F C p C q 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. Z m ord m l irr F α ϕn F F \ {0} X x ĝ ring of integers modulo m multiplicative order of l modulo m minimal polynomial of α over F Euler s totient function of n x X g

3 UNITS IN F q k C p r C q 23 og order of g [ g ] conjugacy class of g G m R m Mm, F GLm, F 3. Preliminaries external direct product of m copies of G external direct sum of m copies of R algebra of all m m matrices over F general linear group of all m m invertible matrices over F The following results will be useful for our investigation. Proposition 3.1. [19, Chapter 1, Proposition 6.16] Let f : R 1 R 2 be a surjective homomorphism of rings. Then f JR 1 JR 2 with equality if ker f JR 1. Remark 3.2. Note that if we add the semi-simplicity of the ring R 2 in the above proposition, then JR 1 ker f. Theorem 3.3. Let E = F ξ/f be a finite separable extension, K be any field extension of F and gx = irr F ξ. If r gx = g i X i=1 as a product of irreducible polynomials g i X K[X], then K F E = r Kξ i i=1 as K-algebras, where ξ i is a root of g i X in an algebraic closure L of K. Proof. For each i, 1 i r, the map λ i : K E Kξ i given by the assignment α, fξ αfξ i α K, fx F [X] is F -bilinear and hence induces a K-algebra homomorphism λ i : K F E Kξ i such that α fξ αfξ i Evidently λ i is onto. Therefore ker λ i is a maximal ideal of K F E for all i, 1 i r and ker λ i + ker λ j = K F E for any i j as 1 g i ξ ker λ i \ ker λ j. It follows from [8, Chapter 5, Theorem 2.2] that the K-algebra homomorphism λ : K F E r Kξ i defined by i=1 λa = λ 1A,..., λ ra A K F E

4 24 N. MAKHIJANI, R. K. SHARMA AND J. B. SRIVASTAVA is onto. Since dim K K F E = dim F E = deg gx = r r deg g i X = dim K Kξ i, i=1 i=1 the proof follows. Theorem 3.4. [23, Theorem 2.21] The distinct automorphisms of F u k over F u are exactly the mappings σ 0,..., σ k 1, defined by σ j α = α uj for α F u k and 0 j k 1. Thus if v = u k, then F v Fu F u m = F v o k m m,k 3.1 as F v -algebras where o k m = ord um 1v = 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 JE F A = E F JA A 2 E F JA = E F A E F JA As a consequence, if A is a finite dimensional F -algebra, then dim E JE F A = dim F JA. 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 the p-regular elements of G, ξ be a primitive s th root of unity over F and T G,F be the multiplicative group consisting of those integers t, taken modulo s, for which ξ ξ t defines an automorphism of F ξ over F. If F = F u, then by Theorem 3.4, where c = ord s u. T G,F = {1, u,, u c 1 } mod s We denote the sum of all conjugates of g G by γ g. Definition 3.6. If g G is a p-regular element, then the cyclotomic F -class of γ g is defined to be the set S F γ g = { γ g t t T G,F } Proposition 3.7. [11, Proposition 1.2] The number of simple components of F G/JF G is equal to the number of cyclotomic F -classes in G.

5 UNITS IN F q k C p r C q 25 Theorem 3.8. [11, Theorem 1.3] Suppose that GalF ξ : F is cyclic. Let w be the number of cyclotomic F -classes in G. If K 1,, K w are the simple components of ZF G/JF G and S 1,, S w are the cyclotomic F -classes of G, then with a suitable re-ordering of indices, S i = [K i : 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 decomposition of the group algebra F q kc n r C q 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 = C n r C q. Then F G/JF G = F 1+ d An ϕd Mq, F d Bn ϕd q where A n = { d d > 1, d n and ord d r = 1 }, B n = { d d > 1, d n and ord d r = q }. Proof. A q-regular conjugacy class of G is either {1} or of the form { } a i if oa i A n [ a i ] = { } a i, a ri,, a rq 1 i if oa i B n showing that if d B n, then there are s d = ϕd/q distinct conjugacy classes in G containing elements of order d, each of size q. Let { } a Im d 1 m s d be the representatives of these classes and C n = { d, m d B n, 1 m s d }. Define the following F -algebra homomorphisms: a φ 1 : F G F by the assignment a 1, b 1. b For each d, m C n, θ m d : F G Mq, F by a ζ Im d ζ Im d r ζ Im d rq 1, b q q q q

6 26 N. MAKHIJANI, R. K. SHARMA AND J. B. SRIVASTAVA Firstly suppose that A n and for each d A n, define the F -algebra homomorphism by the assignment φ d : F G F ϕd a ζ n i d, b 1,, 1 i UZ d }{{} ϕd We claim that if θ : F G F F d An ϕd Mq, F d Bn s d is defined as θ = φ 1 φ d d,m C n θ m d, then dim F ker θ = q ϕd q 1 n 1 n 1 Let X = α j i bj a i ker θ and F j X = α j i Xi F [X] for all j, 0 j q 1. Then a b q 1 q 1 n 1 α j i = 0 θ m d b j F j θ m d a = O d, m C n. F j ζ Im d ri = 0 d, m C n, 0 i, j q 1 Since Φ d X = s d m=1 q 1 X ζ Im d ri d B n, therefore F j X = G j X Φ d X c for some G j X = β j i Xi F [X], where m = n 1 ϕd = ϕd. q 1 q 1 m F j ζ n d i = 0 i UZ d, d A n. F j X = Φ d X gx, for some gx F [X].

7 UNITS IN F q k C p r C q 27 q 1 G j X Φ d X = Φ d X gx Due to degree constraints, we have gx = α Φ d X for some α F. Using a, g1 = 0 and hence gx = 0 q 1 G j X = 0 q 1 β j i = 0 0 i m dim F ker θ = q 11 + m = q 1 Hence the claim follows. 1 + ϕd Evidently if A n = and θ : F G F Mq, F d Bn θ = φ 1 then dim F ker θ = q 1. d,m C n θ m d ϕd q is defined as Thus in either case, θ is onto and hence JF G ker θ showing that θ : F G/JF G θf G defined by θ X + JF G = θx X F G is a well-defined surjective F -algebra homomorphism. As F G/JF G is semi-simple, it follows that F G/JF G = C θf G for the semi-simple F -algebra C = ker θ. But T G,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/JF G is 1 + ϕd ϕd + q

8 28 N. MAKHIJANI, R. K. SHARMA AND J. B. SRIVASTAVA which is same as the number of simple components in θ F G. Hence F G/JF G = θf G and the proof follows. The decomposition may vary if F does not contain a primitive nth root of unity. Theorem 4.2. If u = q k, then F u G/JF u G = F u F u o d x d M q, F u i d y d 4.1 where a o d = ord d u d n, d > 1. b x d = ϕd o d d A n if A n. c for each d B n, i d is the least positive integer such that u i d for some 0 j q 1 and y d = ϕd q i d. r j mod d Proof. Let g G \ {1} be a q-regular element and d = og. The following hold: 1 If A n and d A n, then S Fu γ g = 2 However if d B n, then For any l, l N, 0 j o d 1 { } γ g u j [ g ] = [ a Im d ] for some m, 1 m sd γ g u l = γ g ul [ a ul I m d ] = [ a u l I m d ] u l l r j mod d for some j, 0 j q 1 Therefore S Fu γ g = { } γ g u j 0 j i d 1 Thus there are a x d distinct cyclotomic F u -classes in G, each of order o d, for all d A n if A n. b y d cyclotomic F u -classes, each of order i d, for all d B n.

9 UNITS IN F q k C p r C q 29 In view of the Wedderburn structure theorem and Theorem 3.8, F u G/JF u G 4.2 = F u M n l, F u o d M m j, F u i d for some n l, m j 1. x d l=1 By [10, Theorem 7.9], it follows that Since F u on y d j=1 F u on G/JF u on G = Fu on Fu F u G/JF u G = F u on M n l, F u on Fu F u o d y d j=1 x d l=1 M m j, F u on Fu F u i d contains a primitive nth root of unity, therefore the decomposition of F u on G/JF u on G can be obtained using the previous lemma. As a consequence of equation 3.1 and the uniqueness of Wedderburn decomposition, we have n l, m j = 1 or q. Let κ : F u G F u be the F u -algebra homomorphism, coming from the trivial representation of G and K = ker κ. If A n and d A n, then x d Φ d X = fdx i where f i d X F u[x] is an irreducible polynomial of degree o d for each 1 i x d. i=1 Now for each d, i G n = { d, i d A n, 1 i x d }, let ξd i F u o d be a root of fd ix and κi d : F ug F u o d be an F u -algebra homomorphism obtained from the assignment a ξ i d, b 1 Observe that for all d, i G n, κ i d is onto and hence Ki d = ker κi d is a maximal ideal of F u G. Since it follows that f i da K i d \ K i d d, i, d, i G n and d, i d, i, { K, K i d d, i Gn } is a collection of pairwise co-maximal ideals of F u G. Therefore by Chinese remainder theorem [8, Chapter 5, Theorem 2.2], it follows that F u G K = Fu F u o d d,i G n K i d x d

10 30 N. MAKHIJANI, R. K. SHARMA AND J. B. SRIVASTAVA Using this with Remark 3.2, there exists a surjective F u -algebra homomorphism F u G λ : JF u G M = F u F u o d x d showing that M occurs in the decomposition of F u G/JF u G given in equation 4.2. Recall that n l, m j = 1 or q and 1 + o d x d + q 2 i d y d = 1 + ϕd + q ϕd = n + q 1 ϕd = nq q 1 n ϕd = G dim Fu on JF u on G = G dim Fu JF u G using [10, Theorem 7.9] Therefore F u G/JF u G = F u F u o d x d Via similar arguments the theorem holds true even when A n =. M q, F u i d y d 5. Main result The main result of the paper is as follows. Theorem 5.1. Let u = q k and G = C n r C q where n, q = 1. If ord d r = q for every divisor d> 1 of n, then U F u G = C kq 1 q where i d is the least positive integer such that u i d C u 1 GLq, F u i d y d r j mod d d >1 d n for some 0 j q 1 and y d = ϕd q i d. In particular, U F u C p C q = C kq 1 q C u 1 GL q, F u ip yp

11 UNITS IN F q k C p r C q 31 where p is a prime different from q. Proof. Since b i â ZF u G 0 i q 1, therefore if X F u G, showing that { q 1 α i b i â q 1 q 1 + X α i b â i = 1 α i F u such that q 1 } q 1 α i = 0 JF u G. But from the proof of Theorem 4.2, it follows that dim Fu hence equality. Thus 1 + JF u G = Cq kq 1. Since JF u G ZF u G and UF u G = 1 + JF u G U F u G/JF u G α i = 0, then for any JF u G = q 1 and therefore the proof follows by using Theorem 4.2. Corollary 5.2. Let D 2n be the dihedral group of order 2n, n odd and u = 2 k. Then where o d = ord d u, i d = and y d = ϕd 2i d. U F u D 2n = C2 k C u 1 GL2, F u i d y d o d 2 o d d >1 d n if o d is even and u o d/2 1 mod d otherwise Remark 5.3. This is in accordance with the structure of the unit group of F 2 kd 2n determined by the authors in [24] for odd n. References [1] G. K. Bakshi, S. Gupta and I. B. S. Passi, Semisimple metacyclic group algebras, Proc. Indian Acad. Sci. Math. Sci., , [2] G. K. Bakshi, S. Gupta and I. B. S. Passi, The structure of finite semisimple metacyclic group algebras, J. Ramanujan Math. Soc., , [3] A. Bovdi, The group of units of a group algebra of characteristic p, Publ. Math. Debrecen, ,

12 32 N. MAKHIJANI, R. K. SHARMA AND J. B. SRIVASTAVA [4] V. Bovdi, Group algebras whose group of units is powerful, J. Aust. Math. Soc., , [5] V. Bovdi, L. G. Kovács and S. K. Sehgal, Symmetric units in modular group algebras, Comm. Algebra, , [6] A. A. Bovdi and J. Kurdics, Lie properties of the group algebra and the nilpotency class of the group of units, J. Algebra, , [7] O. Broche and Á. del Río, Wedderburn decomposition of finite group algebras, Finite Fields Appl., , [8] P. M. Cohn, Algebra, Vol. 2, Second edition, John Wiley & Sons, Ltd., Chichester, [9] C. W. Curtis and I. Reiner, Representation Theory of Finite Groups and Associative Algebras, Pure and Applied Mathematics, Vol. XI, Interscience Publishers, a division of John Wiley & Sons, New York-London, [10] C. W. Curtis and I. Reiner, Methods of Representation Theory, Vol. I, with applications to finite groups and orders, Pure and Applied Mathematics, a Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, [11] R. A. Ferraz, Simple components of the center of F G/JF G, Comm. Algebra, , [12] J. Gildea, The structure of the unit group of the group algebra F 3 kc 3 D 6, Comm. Algebra, , [13] J. Gildea, Units of the group algebra F 2 kc 2 D 8, J. Algebra Appl., , [14] J. Gildea, The structure of the unitary units of the group algebra F 2 kd 8, Int. Electron. J. Algebra, , [15] J. Gildea, Units of the group algebra F 5 kc 5 C 4, Int. Electron. J. Algebra, , [16] J. Gildea, The structure of the unit group of the group algebra F 2 ka 4, Czechoslovak Math. J., , [17] P. Hurley and T. Hurley, Codes from zero-divisors and units in group rings, Int. J. Inf. Coding Theory, , [18] B. Hurley and T. Hurley, Group ring cryptography, Int. J. Pure Appl. Math., , [19] G. Karpilovsky, The Jacobson Radical of Group Algebras, North-Holland Mathematics Studies, 135, Notas de Matemtica [Mathematical Notes], 115, North-Holland Publishing Co., Amsterdam, 1987.

13 UNITS IN F q k C p r C q 33 [20] K. Kaur and M. Khan, Structure of the Unit Group of F C p C q, Conference on Groups, Group Rings and Related Topics, [21] K. Kaur and M. Khan, Units in F 2 D 2p, J. Algebra Appl., , 9 pp., DOI: /S [22] M. Khan, R. K. Sharma and J. B. Srivastava, The unit group of F S 4, Acta Math. Hungar., , 2008, [23] R. Lidl and H. Niederreiter, Finite Fields, Encyclopedia of Mathematics and its Applications, 20, Cambridge University Press, Cambridge, [24] N. Makhijani, R. K. Sharma and J. B. Srivastava, Units in F 2 kd 2n, Int. J. Group Theory, , [25] R. Sandling, Units in the modular group algebra of a finite abelian p-group, J. Pure Appl. Algebra, , [26] R. K. Sharma, J. B. Srivastava and M. Khan, The unit group of F A 4, Publ. Math. Debrecen, , [27] I. Van Gelder and G. Olteanu, Finite group algebras of nilpotent groups: a complete set of orthogonal primitive idempotents, Finite Fields Appl., , [28] I. Woungang, S. Misra and S. C. Misra, Selected Topics in Information and Coding Theory, Series on Coding Theory and Cryptology, 7, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, N. Makhijani, R. K. Sharma and J. B. Srivastava Department of Mathematics Indian Institute of Technology New Delhi, India s: nehamakhijani@gmail.com N. Makhijani rksharmaiitd@gmail.com R. K. Sharma jbsrivas@gmail.com J. B. Srivastava

FINITE SEMISIMPLE LOOP ALGEBRAS OF INDECOMPOSABLE RA LOOPS. 1. Introduction

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