ISSN 2148-838Xhttp://dx.doi.org/10.13069/jacodesmath.83854 J. Algebra Comb. Discrete Appl. 3(1) • 1–6 Received: 5 August 2015 Accepted: 20 October 2015 Journal of Algebra Combinatorics Discrete Structures and Applications The unit group of group algebra FqSL(2, Z3) Research Article Swati Maheshwari, R. K. Sharma Abstract: Let Fq be a finite field of characteristic p having q elements, where q = pk and p ≥ 5. Let SL(2, Z3) be the special linear group of 2 × 2 matrices with determinant 1 over Z3. In this note we establish the structure of the unit group of FqSL(2, Z3). 2010 MSC: 16U60, 20C05 Keywords: Group algebra, Unit group, Finite field 1. Introduction Let FG be a group algebra of a finite group G over a field F and U(FG) be the group of units in FG. It is a classical problem to study units and their properties in group ring theory. The case, when G is a finite abelian group, the structure of FG is studied by Perlis and Walker in [14]. In 2006, T. Hurley introduced a correspondence between group ring and certain ring of matrices (see [6]). As an application of units of a group ring, T. Hurley gave a method to construct convolutional codes from units in group ring (see [7]). A lot of work has been done for finding the algebraic structure of the unit group U(FG) of a group algebra FG, when G is a finite non-abelian group. Here we are providing some literature survey for the same. For dihedral groups, the structure of the unit group U(FG) over a finite field F is discussed in [1, 4, 10, 12]. J. Gildea et.al. (see [3]) and R. K. Sharma et.al. (see [15]) have given the structure of the unit group U(FG), where G is alternating group A4. Unit group of algebra of circulant matrix has been discussed in [11, 17]. The unit group of group algebras of some non-abelian groups with small orders are established in [16, 18, 19]). In this article, we are interested in studying the structure of the unit group of FqSL(2,Z3) over a finite field of characteristic greater than 3. This work was supported by IIT Delhi, India through GATE Senior Research Fellowship. Swati Maheshwari (Corresponding Author), R. K. Sharma; Department of Mathematics, Indian Institute of Technology Delhi, India (email: swatimahesh88@gmail.com, rksharmaiitd@gmail.com). 1 S. Maheshwari, R. K. Sharma / J. Algebra Comb. Discrete Appl. 3(1) (2016) 1–6 2. Preliminaries The following results provide useful information about the decomposition of A/J(A), where A = FG, J(A) be its Jacobson radical and F being a field of characteristic p. For basic definitions and results, we refer to [13]. We briefly introduce some definitions and notations those will be needed subsequently. Definition 2.1. An element g ∈ G is said to be p-regular if p - o(g). Let s be the l.c.m. of the orders of the p-regular elements of G, ζ be a primitive s-th root of unity over F. Then TG,F be the multiplicative group consisting of those integers t, taken modulo s, for which ζ 7→ ζt defines an automorphism of F(ζ) over F. That is, TG,F is Gal(F(ζ)/F) seen as a subgroup of U(Zs). Note that if u is a power of a prime such that (u,s) = 1 and c = ords (u) is the multiplicative order of u modulo s, then TG,Fu = {1,u, . . . ,u c−1} mod s and Fu(ζ) ∼= Fuc follow using [8, Theorem 2.21]. Definition 2.2. If g ∈ G is a p-regular element, then the sum of all conjugates of g ∈ G is denoted by γg and the cyclotomic F-class of g is defined to be the set SF(γg) = {γgt | t ∈ TG,F}. Proposition 2.3. [2, Theorem 1.2] The number of simple components of FG/J(FG) is equal to the number of cyclotomic F-classes in G. Theorem 2.4. [2, Theorem 1.3] Suppose that Gal(F(ζ)/F) is cyclic. Let w be the number of cyclotomic F-classes in G. If K1,K2, . . . ,Kw are the simple components of Z(FG/J(FG)) and S1,S2, . . . ,Sw are the cyclotomic F-classes of G, then with a suitable re-ordering of indices, | Si |= [Ki : F]. Lemma 2.5. [9, Observation 2.2.1, p.22] Let B1,B2 be two finite dimensional F-algebras such that B2 is semisimple. If f : B1 → B2 is an onto homomorphism of F-algebras, then there exists a semisimple F-algebra ` such that B1/J(B1) ∼= `⊕B2. Throughout this article, G = SL(2,Z3). Fq is a field of characteristic p, where q = pk and k is a positive integer. The conjugacy class of g ∈ G is denoted by [g]. 3. Main result We shall use the presentation of G given in [5], 〈a,b | a3,b4,(ab)3 = b2,(a2b)6〉 where a = [ 1 0 1 1 ] and b = [ 0 1 −1 0 ] . We can see that G has 7 conjugacy classes as follows: 2 S. Maheshwari, R. K. Sharma / J. Algebra Comb. Discrete Appl. 3(1) (2016) 1–6 representative elements in the class order of element [a] a,(ba)4,(ab)4,b−1ab 3 [a−1] a−1,(ba)2,(ab)2,aba 3 [b] b,b−1,a2ba,aba2,ab−1a2,a2b−1a 4 [b2] b2 2 [ab] ab,ba,a2ba2,ab2 6 [(ab)−1] (ab)−1,a2b−1,ab−1a,a2b2 6 We have (p, |G|) = 1 and so J(FpkG) = 0. Further, we discuss the decomposition of FpkG. Theorem 3.1. Let Fq be a finite field of characteristic p, where p ≥ 5. Then the Wedderburn decompo- sition of FqG is given by condition on k FqG k is even F3q ⊕M(2,Fq)3 ⊕M(3,Fq) k is odd p ≡ 1 mod 3 and p ≡±1mod 4 F3q ⊕M(2,Fq)3 ⊕M(3,Fq) k is odd p ≡−1 mod 3 and p ≡±1mod 4 Fq ⊕Fq2 ⊕M(2,Fq)⊕M(2,Fq2)⊕M(3,Fq) Proof. Since FqG is semisimple, so it has the Wedderburn decomposition which is given by FqG ∼= ⊕ri=1M(ni,Fi), where for each i,ni ≥ 1and Fi is a finite extension of Fq. By using Lemma 2.5, we have FqG ∼= Fq ⊕r−1i=1 M(ni,Fi). (1) Further, we find ni’s and Fi’s. Since | G |= 24, hence any element g ∈ G is a p- regular element. For finding cyclotomic Fq - classes of G, first we assume that k is even. We have pk ≡ 1 mod 4 and pk ≡ 1 mod 3. Then by Chinese remainder theorem pk ≡ 1 mod 12. By using above observation, we have SFq (γg) = {γg} and | SFq (γg) |= 1. Therefore by using Equation (1), Proposition 2.3 and Theorem 2.4, we have FqG ∼= Fq ⊕6i=1 M(ni,Fq) 3 S. Maheshwari, R. K. Sharma / J. Algebra Comb. Discrete Appl. 3(1) (2016) 1–6 for some ni ≥ 1. As dimension of FqG is 24, we get 6∑ i=1 n2i = 23. Using above equality, 1 ≤ ni ≤ 3. Clearly any ni = nj = 3 for 1 ≤ i 6= j ≤ 3 not possible. So the only possible choice for ni’s is n1 = n2 = 1,n3 = n4 = n5 = 2 and n6 = 3. Therefore the decomposition FqG is given by FqG ∼= F3q ⊕M(2,Fq)3 ⊕M(3,Fq). Now we consider the case when k is odd. We shall discuss this case into two parts 1. p ≡ 1 mod 3 and p ≡±1 mod 4 2. p ≡−1 mod 3 and p ≡±1 mod 4 Case 1. Suppose k is odd with p ≡ 1 mod 3 and p ≡±1 mod 4. Observe that pk ≡ p mod 4 and pk ≡ p mod 3. Then by Chinese remainder theorem pk ≡ p mod 12. Since [b] = [b−1]. We have SFq (γg) = {γg}. Hence ni’s and Fi’s are same as above. So the decomposition of FqG is given by FqG ∼= F3q ⊕M(2,Fq) 3 ⊕M(3,Fq). Case 2. Suppose k is odd with p ≡−1 mod 3 and p ≡±1 mod 4. Using the observation in case 1, we have pk ≡ p mod 12. SFq(γb) = {γb},SFq(γb2) = {γb2}, SFq(γa) = {γa,γa−1} and SFq(γab) = {γab,γ(ab)−1}. Therefore by using Equation (1), Proposition 2.3 and Theorem 2.4, we have FqG ∼= Fq ⊕M(n1,Fq)⊕M(n2,Fq)⊕M(n3,Fq2)⊕M(n4,Fq2) for some ni ≥ 1. As dimension of FqG is 24, we get n21 + n 2 2 + 2n 2 3 + 2n 2 4 = 23 and hence, 1 ≤ ni ≤ 3, ∀1 ≤ i ≤ 4. Clearly n3 and n4 can not be equal to 3. So the only possible choice for ni’s is n1 = 2,n2 = 3,n3 = 1,n4 = 2. Therefore the decomposition of FqG is given by FqG ∼= Fq ⊕Fq2 ⊕M(2,Fq)⊕M(2,Fq2)⊕M(3,Fq). 4 S. Maheshwari, R. K. Sharma / J. Algebra Comb. Discrete Appl. 3(1) (2016) 1–6 Corollary 3.2. Let q = pk,where p ≥ 5 is a prime. Then the structure of U(FqG) is given by condition on k U(FqG) k is even C3q−1 ⊕GL(2,Fq)3 ⊕GL(3,Fq) k is odd p ≡ 1 mod 3 and p ≡±1mod 4 C3q−1 ⊕GL(2,Fq)3 ⊕GL(3,Fq) k is odd p ≡−1 mod 3,±1mod 4 Cq−1 ⊕Cq2−1 ⊕GL(2,Fq)⊕GL(2,Fq2)⊕GL(3,Fq) Proof. It follows by the fact that, if R and S are two rings then U(R⊕S) = U(R)⊕U(S). References [1] L. Creedon, J. Gildea, The structure of the unit group of the group algebra F2kD8, Canad. Math. Bull. 54(2) (2011) 237–243. [2] R. A. Ferraz, Simple components of the center of FG/J(FG), Comm. Algebra 36(9) (2008) 3191–3199. [3] J. Gildea, The structure of the unit group of the group algebra Fk2A4, Czechoslovak Math. J. 61(2) (2011) 531–539. [4] J. Gildea, F. Monaghan, Units of some group algebras of groups of order 12 over any finite field of characteristic 3, Algebra Discrete Math. 11(1) (2011) 46–58. [5] P. R. Helm, A presentation for SL(2,Zpr ), Comm. Algebra 10(15) (1982) 1683–1688. [6] T. Hurley, Group rings and ring of matrices, Int. J. Pure Appl. 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