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Southeast Asian Bulletin of (2019) 43: 217–224 Southeast Asian Bulletin of Mathematics c SEAMS. 2019

A Note on Presentation of General Linear Groups over a Finite

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 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 n.

Keywords: Lie regular units; ; 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 of n × n matrices with 1. We know that GL(n, F) can be written as a , GL(n, F)= SL(n, F) oF∗, where F∗ denotes the 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 . 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 (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 of GL(n, F) is dependent on F. It motivates us to find a pre- sentation of GL(n, F) with fix 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 group of the ring R. Suppose G be a group then o(G) denotes the 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 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 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 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- 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. We have some observations, n (i) If a1a2 = ak0 , then akak+1 = ak0+(k−1) for 1 ≤ k, k0 ≤ 2 − 1 and k0 is such that α + α2 = αk0 . We shall show this by induction, for k = 1 it

holds. Assume for k = m, amam+1 = ak0+(m−1), from here we get

bab(bc)−1 = (bc)1−k0 bab(bc)k0−2bab. (4.1)

Let k = m +1. Then

−1−m −1 m+2 am+1am+2 =(bc) bab(bc) bab(bc) =(bc)−m−k0 bab(bc)m+k0 using (4.1)

=ak0+m.

(ii) baib = bi. 2 (iii) (abi) = 1. The statement holds by using the relation aaia = (bab)bi(bab) and the observation (1). −1 (iv) caic = bi−1. As relations in the group H can be obtained by using above observations and relations in the group G. Which implies that the group H is a subgroup of G, infact it is a of the group G. Corollary 2.4 and Theorem 2.5, give that the group H is isomorphic to SL(2, F2n ). Now consider the G/H, we see that

n G/H = hc | c2 −1i. and o(G)= o(GL(2, F2n )). This proves that G =∼ GL(2, F2n ).

In F2, we see that α = 1, so the element b is same as the element c. Hence we have a remark for presentation of GL(2, F2).

Remark 4.2. A peresentation of GL(2, F2) is

ha,b | a2,b2, (ab)3i.

In the following corollary we are giving a presentation of GL(2, F2n ), where 2 ≤ n ≤ 7,n 6= 5. In this case, we have a primitive , 1 + x + xn (see [9]) and hence α + α2 = αn+1. 222 S. Maheshwari and R.K. Sharma

Corollary 4.3. A presentation of GL(2, F2n ) is

2 2 2(2n−1) 2 3 2n−1 ha,b,c | a ,b ,c ,c ∈ center, (ab) , (bc) ,aaia = (bab)bi(bab),

a1a2 = an+1i.

5. Presentation of GL(2, Fpn )

Theorem 5.1. Let p be an odd prime, q = pn for n ≥ 1 and α is a primitive element of Fq. Let 1 0 0 1 0 α a = ,b = ,c = −1 −1 1 0 1 0

−1 −1 −i q i j q −j and ai = (bc) (b(bc) 2 ab)(bc) , bi = (bc) ((bc) 2 a)(bc) for 1 ≤ i, j ≤ q−1. Then a presentation of GL(2, Fq) is

q−1 ha,b,c | a2,b2,c2(q−1),c2 ∈ center, (ab)3, ((bc) 2 a)p, (bc)q−1, 0 i 0 a1a2 = a2a1 = ak0 , a1 = aki0 (2 ≤ i ≤ p − 1), bq−1bi = bibq−1, q−1 −1 −1 2 −1 aq−1bq−1aq−1 = b(bc) , aaia = aq−1biaq−1, −1 −1 −1 −1 (aq−1bq−1aq−1)bm(aq−1bq−1aq−1) = am i,

2 k0 0 k 0 where k0 and ki0 are such that α+α = α , i α = α i and 1 ≤ k0, ki0 ,m ≤ q−1.

2 2 Proof. Let α + α be not an invertible element in Fq. Then α + α = 0, it provides that α = −1. This is the possible case when q = 3. We will discuss this 2 case separately. Hence α + α is invertible in Fq, q 6= 3, so there exist k0 such 2 k0 0 that α +α = α . Since i and α are non-zero elements of Fq. Hence there exist 0 k 0 0 ki0 such that ı α = α i as ı α is a non-zero element. Suppose G is a subgroup of GL(2, Fq) generated by a,b,c and having presentation

q−1 ha,b,c | a2, b2, c2(q−1), c2 ∈ center, (ab)3, ((bc) 2 a)p, (bc)q−1, 0 i a1a2 = a2a1 = ak0 , a1 = aki0 , bq−1bi = bibq−1, q−1 −1 −1 2 −1 aq−1bq−1aq−1 = b(bc) , aaia = aq−1biaq−1, −1 −1 −1 −1 (aq−1bq−1aq−1)bm(aq−1bq−1aq−1) = am i. First we shall show that group G is finite. Consider a subgroup H of G which is given by

p p −1 −1 −1 H = hai, bj | ai ,bj , aiaj = ak, bibj = bk, (aibq−1−iai)bj (aibq−1−iai) = −1 −1 −1 −1 −1 a2i+j , (biaq−1−ibi)aj (biaq−1−ibi) = b2i+j i, where k ∈ N is such that αi +αj = αk, 1 ≤ k ≤ q−1. We have some observations,

(i) baib = bi. Presentation of General Linear Groups over a Finite Field 223

2 (ii) (abi) = 1. The statement holds by using relation bq−1bi = bibq−1. −1 (iii) caic = bi−1. By using the above observations and relations in the group G, we see that the relations in the group H can be obtained. Hence H is a subgroup of the group G, infact normal subgroup of G. Corollary 2.4 and Theorem 2.5, give that the group H is isomorphic to SL(2, Fq). Now we consider the quotient group G/H. q−1 Case 1. When 2 is even we have, G/H = hc | cq−1i. q−1 Case 2. When 2 is odd we have,

q−1 G/H = hb,c | b2,c 2 ,bc = cbi.

In both cases we see that o(G/H)= q − 1. This proves that G =∼ GL(2, Fq).

Corollary 5.2. When q =3, a presentation of GL(2, Fq) is 2 2 4 2 3 p 2 2 ha,b,c | a ,b ,c ,c ∈ center, (ab) , (bca) , (bc) , a1 = a2, bq−1bi = bibq−1, q−1 −1 −1 2 −1 aq−1bq−1aq−1 = b(bc) , aaia = aq−1biaq−1, −1 −1 −1 −1 (aq−1bq−1aq−1)bm(aq−1bq−1aq−1) = am i, where 1 ≤ m ≤ 2.

In the following corollary we are giving a presentation of GL(2, F9). In this case, we have a primitive polynomial, 2 + x + x2.

Corollary 5.3. A presentation of GL(2, F9) is ha,b,c | a2,b2,c16,c2 ∈ center, (ab)3, ((bc)4a)3, (bc)8, 2 −1 −1 4 a1a2 = a2a1 = a8, a1 = a5, b8bi = bib8, a8 b8a8 = b(bc) , −1 −1 −1 −1 −1 aaia = a8 bia8, (a8b8 a8)bm(a8b8 a8) = am i.

Thus we have found presentations for GL(2, F2n ) and GL(2, Fpn ) in terms of Lie regular units. The exciting thing about these generators is that these ele- 0 α 1 0 ments have a fixed form for some α ∈ Fq\{0}, except and 1 0 1 1 1 0 . −1 −1 All results are verified with MAGMA software. References

[1] G. Chiaselotti, Some presentations for the special linear groups on finite fields, Annali di Mat. 180 (2001) 359–372. 224 S. Maheshwari and R.K. Sharma

[2] T.A. Francis, Presentations of the special and general linear groups, J. Algebra 169 (1994) 943–964. [3] S.M. Green, Generators and relations for the special linear group over a , Proc. Am. Math. Soc. 62 (2) (1977) 229–232. [4] P. Kanwar, R.K. Sharma, P. Yadav, Lie regular generators of general linear groups II, Int. Electron. J. Algebra 13 (2013) 91–108. [5] J. Konieczny, The semigroup is generated by regular boolean matrices, Southeast Asian Bull. Math. 25 (2002) 627–641. [6] W. Magnus, A. Karrass, D. Solitar, Combinatorial ; Presentations of Groups in Terms of Generators and Relations, Dover Publications, Inc., New York, 1976. [7] S. Maheshwari and R.K. Sharma, Lie regular generators of general linear group GL(4, Zn), Serdica Math. J. 42 (2016) 211–220. [8] R.K. Sharma, P. Yadav, P. Kanwar, Lie regular generators of general linear groups, Commun. Algebra 40 (4) (2012) 1304–1315. [9] W. Stahne, Binary primitive , Math. Comput. 27 (124) (1973) 977– 980.