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A Note on Presentation of General Linear Groups Over a Finite Field

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A Note on Presentation of General Linear Groups Over a Finite Field Southeast Asian Bulletin of Mathematics (2019) 43: 217–224 Southeast Asian Bulletin of Mathematics c SEAMS. 2019 A Note on Presentation of General Linear Groups over a Finite Field Swati Maheshwari and R. K. Sharma Department of Mathematics, Indian Institute of Technology Delhi, New Delhi, India Email: [email protected]; [email protected] Received 22 September 2016 Accepted 20 June 2018 Communicated by J.M.P. Balmaceda AMS Mathematics Subject Classification(2000): 20F05, 16U60, 20H25 Abstract. In this article we have given Lie regular generators of linear group GL(2, Fq), n where Fq is a finite field with q = p elements. Using these generators we have obtained presentations of the linear groups GL(2, F2n ) and GL(2, Fpn ) for each positive integer n. Keywords: Lie regular units; General linear group; Presentation of a group; Finite field. 1. Introduction Suppose F is a finite field and GL(n, F) is the general linear the group of n × n invertible matrices and SL(n, F) is special linear group of n × n matrices with determinant 1. We know that GL(n, F) can be written as a semidirect product, GL(n, F)= SL(n, F) oF∗, where F∗ denotes the multiplicative group of F. Let H and K be two groups having presentations H = hX | Ri and K = hY | Si, then a presentation of semidirect product of H and K is given by, −1 H oη K = hX, Y | R,S,xyx = η(y)(x) ∀x ∈ X,y ∈ Y i, where η : K → Aut(H) is a group homomorphism. Now we summarize some literature survey related to the presentation of groups. In 1977, S.M. Green es- tablished a presentation of SL(n, R) for, n ≥ 3 (see [3]) and in 1994, T.A. Fransis has found a presentation of GL(n, R), where R is a division ring (see [2]). In case R is a field, T.A. Fransis has also provided a presentation of SL(n, R). 218 S. Maheshwari and R.K. Sharma Generators for the semigroup has been provided by J. Konieczny (see [5]). A presentation of SL(n, F) has also been given by G. Chiaselotti (see [1]). We have seen that a presentation of SL(n, F) has been found, so we can always find one presentation of GL(n, F) using semidirect product. In that case, cardinality of the generating set of GL(n, F) is dependent on F. It motivates us to find a pre- sentation of GL(n, F) with fix number of generators. In this article, we establish a presentation of GL(2, Fq) with fix number of generators, where generators are Lie regular units. These elements have been first introduced by R. K. Sharma et al. in 2012 (see [8, 4]). A generating set for GL(4, Zn) has been found by S. Maheshwari and R. K. Sharma in 2016 (see [7]). This article is arranged in the following way: Section 2, contains some basic definitions and well known results. In Section 3, we have shown that Lie regular units generate GL(2, Fq) (see The- orems 3.1 and 3.2). In the last Section, we have established a presentation for GL(2, F2n ) and GL(2, Fpn ) (see Theorems 3.3 and 3.2) with generating set of cardinality 3. Throughout this paper, φ denotes the Euler’s totient function and U(R) denotes the unit group of the ring R. Suppose G be a group then o(G) denotes the order of the group G. 2. Preliminaries Here we record some well known results and basic definitions, which we shall use frequently in this note. F F o(GL(2, q)) Lemma 2.1. The order of the special linear group SL(2, q) is q−1 . Definition 2.2. An element ‘a’ of a ring R is said to be Lie regular if a = [e,u]= eu − ue, where e is an idempotent of R and u is a unit of R. Further, a unit in R is said to be Lie regular unit if it is Lie regular as an element of R. In the following lemma eij (r) for 1 ≤ i, j ≤ n and r ∈ Fq denotes elementary matrix of the form eij = I + rij , where ij denotes the matrix with 1 on the (i,j)-th position and 0 elsewhere and I is the n × n identity matrix. Lemma 2.3. [2, p. 944] Suppose Fq is a finite field. Then SL(n, Fq) has a presentation with generators eij (r) and relations: (i) eij (r)eij (s)= eij (r + s), (ii) [eij (r),ekl(s)] = 1 if i 6= l, j 6= k, (iii) [eij (r),ejk (s)] = eik(rs) if i, j and k are distinct, and −1 −1 F∗ (iv) eeji(r)e = eij (−trt) for e = eij (t)eji(−t )eij (t),t ∈ q. n Let α be a primitive element of Fq, where q = p and 1 ≤ i, j ≤ q − 1. For convenience, we rewrite the above presentation in new symbols as follows: Presentation of General Linear Groups over a Finite Field 219 Corollary 2.4. Let 1 αi 1 0 ai = ,bj = . 0 1 αj 1 Then SL(2, Fpn ) is generated by ai and bj with these relations: p p (i) ai =1 and bj =1, i j k (ii) for i 6= j, we have aiaj = ak, bibj = bk, where k is such that α +α = α , and 1 ≤ k ≤ q − 1, −1 −1 −1 −1 (iii) (aibq−1−iai)(bj )(aibq−1−iai) = a2i+j , −1 −1 −1 −1 (iv) (biaq−1−ibi)(aj )(biaq−1−ibi) = b2i+j . Theorem 2.5. [6, Theorem 1.1] Two groups having same presentation are iso- morphic. 3. Lie Regular Generators of General Linear Groups Observe that the element 1 0 0 −1 0 1 a = = , −1 −1 0 1 −1 −1 0 k is a Lie regular unit in M (Fq) and b = , where k is invertible in Fq, is 2 1 0 also a Lie regular unit in M2(Fq) (see [8, Proposition 2.14]). F F∗ Theorem 3.1. Suppose 2n is a finite field and α ∈ 2n is a primitive element n i.e. o(α)=2 − 1. Then the linear group GL(2, F2n ) is generated by Lie regular units a,b and c, where 1 0 0 1 0 α a = ,b = ,c = . 1 1 1 0 1 0 Proof. Let G be a subgroup of GL(2, F2n ) generated by a,b and c. Set i −i i 1 α j −j 1 0 ai = (bc) (bab)(bc) = ,bj = (bc) a(bc) = , 0 1 αj 1 n where 1 ≤ i, j ≤ 2 − 1. By using Corollary 2.4, ai and bj generate SL(2, Fq). 1 0 Let x = bc = , then order of x is 2n − 1. Consider a subgroup of G, 0 α n H = hx | x2 −1i. We see that H ∩ SL(2, F2n )= {I2}, thus 220 S. Maheshwari and R.K. Sharma n o(HSL(2, F2n )) = (2 − 1)o(SL(2, F2n )). n Since HSL(2, F2n ) ⊆ G ≤ GL(2, F2n ) and by lemma 2.1, o(GL(2, F2n )) = (2 − 1)o(SL(2, F2n )), as a consequence, we have HSL(2, F2n )= GL(2, F2n ). n Theorem 3.2. Suppose Fq is a finite field with q = p , where p is an odd prime F∗ and α ∈ q is a primitive element i.e. o(α) = q − 1. Then the linear group GL(2, Fq) is generated by Lie regular units a,b and c, where 1 0 0 1 0 α a = ,b = ,c = . −1 −1 1 0 1 0 Proof. Let G be a subgroup of GL(2, Fq) generated by a,b and c. Set q−1 i −i 2 i 1 α ai = (bc) (b(bc) ab)(bc) = , 0 1 q−1 j 2 −j 1 0 bj = (bc) ((bc) a)(bc) = , αi 1 where 1 ≤ i, j, ≤ q − 1. By using Corollary 2.4, ai and bj generate SL(2, Fq). Remaining proof is same as proof of Theorem . 4. Presentation of GL(2, F2n ) In the following theorem we assume that 1 ≤ i, j ≤ 2n − 1 and n> 1. Theorem 4.1. Let 1 0 0 1 0 α a = ,b = ,c = , 1 1 1 0 1 0 where α is a primitive element of F2n . Then a presentation of GL(2, F2n ) is n n ha,b,c | a2,b2,c2(2 −1),c2 ∈ center, (ab)3, (bc)2 −1, aaia = (bab)bi(bab) ,a1a2 = ak0 i, 2 k0 where k0 ∈ N is such that α + α = α and i −i i 1 α j −j 1 0 ai = (bc) (bab)(bc) = ,bj = (bc) a(bc) = . 0 1 αj 1 2 Proof. α + α is non-zero element in F2n for n> 1. So there exist k0 such that α + α2 = αk0 . Let G be a group generated by a,b,c and having presentation, n n ha,b,c | a2,b2,c2(2 −1),c2 ∈ center, (ab)3, (bc)2 −1, aaia = (bab)bi(bab),a1a2 = ak0 i. Presentation of General Linear Groups over a Finite Field 221 First, we shall show that G is finite. Consider a group H having the following presentation 2 2 −1 H = hai, bj | ai ,bj ,aiaj = ak,bibj = bk, aib2n−1−iaibj (aib2n−1−iai) = a2i+j , −1 −1 −1 −1 (biaq−1−ibi)(aj )(biaq−1−ibi) = b2i+j i, where k ∈ N is such that αi + αj = αk.
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