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Presentations on standard generators for classical groups

For each family of finite classical groups, and their associated simple quotients, we provide an explicit presentation on a specific generating set of size at most 8. Since there exist efficient algorithms to construct this generating set in any copy of the group, our presentations can be used to verify claimed isomorphisms between representations of the classical group. The presentations are available in Magma.


Introduction
The finite simple groups of Lie type play a fundamental role in many areas of mathematics. Various presentations of these groups satisfying specific requirements are known. Steinberg [26,27] gave the first presentations. An important variation is that of Curtis-Steinberg-Tits (see [12,Theorem 2.9.3]). Motivated in part by a conjecture of [3], Babai et al. [4] provided presentations on a reduced set of Curtis-Steinberg-Tits generators. Guralnick et al. [13,14] provide short presentations (defined below) for every finite simple group of Lie type except the Ree groups 2 G 2 (q). In this paper, we focus on the classical groups. For each family of finite classical groups, and their associated simple quotients, we provide an explicit presentation on a specific generating set of size at most 8. The most important feature of our work is that there exist efficient algorithms to construct this generating set in any copy of the group, so our presentations can be used in practice to verify claimed isomorphisms between representations of the classical group. Encodings of the resulting presentations are available in Magma [2].
The principal motivation for our work is the matrix group recognition project [23]; it aims to develop high-quality algorithms to compute efficiently with an arbitrary subgroup G of GL(m, K), where K is a finite field and G is specified by a generating set Y . The CompositionTree algorithm [1] constructs a composition series for G, and facilitates membership testing in G and writing elements of G as words in Y . Additional functionality building on the data structure provided by CompositionTree includes, for example, the computation of chief series, the soluble radical, and Sylow subgroups. A highly optimised implementation of this machinery is available in Magma.
Central to that work is a requirement to produce a concrete isomorphism between an arbitrary representation of a quasisimple group and (a central quotient of) its standard copy, a specific representation of the group. The standard copy of A n is the group acting on n symbols. Our principal focus here is on the classical groups: namely, SL(d, q), Sp(2n, q), SU(d, q), Ω + (2n, q), Ω(2n + 1, q), and Ω − (2n, q), where d ≥ 2, n ≥ 1 and q is a prime power; the standard copy of each is a natural representation of the group preserving a designated classical form. The standard copy of a quasisimple exceptional group of Lie type over GF(q) is defined in [15]: a specific faithful irreducible representation of minimal dimension of the simply connected group.
Let C be the standard copy of a quasisimple group and let H = X ≤ GL(m, K) be isomorphic to C/Z where Z is central in C. Informally, a constructive recognition algorithm for H constructs an isomorphism between H and C/Z and exploits this isomorphism to write an arbitrary element of H as a word in its generators X. For a more formal definition, see [24, p. 192].
We can realise such an algorithm as follows. For each quasisimple group C, we define a specific sequence of standard generators S. The first task is to construct, as words in X, a sequence S ′ in H such that S → S ′ induces an isomorphism from C/Z to H. The second task is to solve the constructive membership problem for H with respect to S ′ : namely, express any given h ∈ H as a word in S ′ , and so as a word in X. Now the isomorphism ϕ : H → C/Z that maps S ′ to S is constructive: any given h ∈ H may be written as a word w(S ′ ) in S ′ , and the image ϕ(h) is immediately determined by the corresponding word in S. Similarly, the inverse of ϕ is constructive.
Standard generators have been defined and associated constructive recognition algorithms developed for all of the quasisimple groups. The standard generators of A n are a 3-cycle and an (n−1)-or n-cycle depending on the parity of n; for a recognition algorithm, see [17]. For exceptional groups of rank at least 2, see [8,21]. For the sporadic simple groups, see [29,30]. For the classical groups, see [10,11,19]; we use algorithms of Costi [9] and Schneider to solve the constructive membership problem. In all cases, the algorithms demonstrate both excellent theoretical and practical performance. For example, those in [10,19] have complexity O(r 4 log q) where the classical group has rank r. Implementations are available in Magma.
All existing constructive recognition algorithms employ random selection in H and are Monte Carlo; consequently, there is a small probability of error in the claimed isomorphism between H = X ≤ GL(m, K) and C/Z. Presentations on standard generators for (central quotients of) quasisimple groups play a fundamental role: namely, they permit us to verify the correctness of this isomorphism. Assume we know a presentation {S | R} for C/Z. The standard generators S ′ are constructed as words in X, so demonstrating that X ≥ S ′ ; we use the constructive membership algorithm to write the input generators X as words in S ′ , so demonstrating that X ≤ S ′ ; finally, if the relators in R evaluate to the identity on S ′ , then we have verified the claimed isomorphism.
As part of CompositionTree, we must often construct both a homomorphism θ from G ≤ GL(m, K) onto the simple quotient of a specified classical group C, and its associated kernel. We apply (a natural generalisation of) a constructive recognition algorithm to G, so constructing S ′ ; now evaluating R on S ′ provides normal generators for ker θ. More generally, if we construct a composition tree for an arbitrary group G = Y ≤ GL(m, K), have a constructive membership test for each of its composition factors, and know a presentation on the standard generators of each factor, then, as we outline in [1,Section 4], we can construct a presentation for G on Y ; this allows us to verify the correctness of the composition tree constructed for G.
Presentations on standard generators for A n are well known. For exceptional groups of rank at least 2, the requisite presentations are listed in [21]; for (most of) the sporadic simple groups, they are listed in the Atlas [30]. In this paper, we address the missing case by listing explicit presentations on standard generators for the classical groups and their central quotients.
A more detailed account of the background and context of this work appears in the survey [23].
1.1. The main result. Our motivations dictate the following desiderata for presentations of classical groups.
(1) The presentation is correct. (2) The presentation can be used to verify the claimed isomorphism.
(3) The number of relators is small. (4) The presentation length is "short".
The requirement that the presentation is correct may seem obvious, but it is not easy to achieve. We have concentrated on producing presentations whose correctness is easy to verify.
The second is realised by providing a presentation on the standard generators of the group.
The desire for a small number of relators arises because we may have a group G that maps onto a classical group C, and a normal subgroup K of G contained in the kernel of the homomorphism of G onto C. To verify that K is the full kernel, we must test that a preimage of each relator belongs to K.
We seek a "short" presentation to minimise the cost of isomorphism verification. Babai and Szemerédi [3] define the length of a presentation {X | R} to be the number of symbols required to write down the presentation. Typically, two metrics are applied to measure the length. One is word length: |X| plus the sum of the lengths of the words in R as strings in the alphabet X ∪ X −1 . A second is bit-length: |X| plus the total number of bits required to encode the words in R where all exponents are encoded as binary strings. As discussed in [13], we can readily convert a presentation having bit-length N into a presentation having word length 4N . As we show in Section 1.4, for our applications the critical metric is bit-length.
We now state the main result, observing that it satisfies these desiderata. Each classical group has at most eight standard generators, as defined in [10,19]. Theorem 1.1. Every classical group of rank r defined over GF(q) has a presentation on its standard generators with O(r) relations and total bit-length O(r + log q).
For each family of classical groups, and their central quotients, we exhibit such a presentation. Sometimes, for the convenience of theoretical arguments, we list a presentation on a generating set which includes elements obtained via identified elementary operations from the standard generators.
Our presentation for a classical group C of rank at least 2 on its standard generators usually includes one for the normaliser of a maximal torus in a split BN pair, and so for S n , the symmetric group of relevant degree n, or the corresponding group of signed permutation matrices of determinant 1. Our generators for S n in its action on n symbols are a 2-cycle and n-cycle, and every presentation for S n on this generating set has word length O(n) (see [14, A2]). At most 56 additional relations are used to define C.
The structure of the paper is the following. In Section 2 we provide presentations for S n and for the group of signed permutation matrices of degree n and determinant 1. In Sections 3 and 4 we discuss important base cases: SL(2, q) and SU (3, q). In subsequent sections, we provide presentations for each of SL(d, q), Sp(2n, q), SU(d, q), Ω + (2n, q), Ω(2n + 1, q), and Ω − (2n, q).
The presentations and corresponding matrix generators are available [20] in Magma. As we report in Section 12, we used these extensively to verify the correctness of this work.
1.2. Related work. Babai and Szemerédi [3] formulated the Short Presentation Conjecture: there exists a constant c such that every finite simple group G has a presentation of bit-length O(log c |G|). The results of [4,16,28] establish this conjecture with c = 2 for all finite simple groups, with the possible exception of the Ree groups 2 G 2 (q).
The conjecture was motivated by potential complexity applications to questions about matrix groups defined over finite fields (see [3] for details); its proof also provided verification for the first constructive recognition algorithms for classical groups, developed by Kantor and Seress [18].
In two remarkable papers, Guralnick et al. [13,14] establish much more. We summarise just two of their results. Theorem 1.2. Every non-abelian finite simple group of rank r defined over GF(q), with the possible exception of the Ree groups 2 G 2 (q), has a presentation with a bounded number of generators and relations and word length O(log r +log q). Every finite quasisimple group of rank r defined over GF(q), with the possible exception of the Ree groups 2 G 2 (q), has a presentation with at most 9 generators, 49 relations, and bit-length O(log r + log q).
Word and bit-length O(log r + log q) are both best possible, and such presentations are short.
Rarely do we know explicit words (short or otherwise) to express standard generators in terms of the generators used in [14], or vice versa, so it is not feasible to convert their presentations to ones on standard generators. One significant obstruction is that sometimes generators used there are identified only by specifying properties they must satisfy. Nor do constructive recognition algorithms employing their generators exist for classical groups, so these presentations cannot be used directly to verify the necessary isomorphisms. Theorem 10.1 of [14] illustrates an additional concern: it employs a generator σ specified only up to certain properties. This element does not exist in Ω + (2n, q); in private communication in 2012, the authors of [14] fixed the error. Thus, while the results of [13,14] provide spectacular answers to long-standing challenging problems, we believe that our presentations are necessary for our significant algorithmic application.
1.3. Our approach. Our general strategy to construct a presentation for a classical group C is straight-forward.
Our generating set usually contains generators for the normaliser N of a maximal torus in a split BN pair, and one root element of each type. For example, we take ( 1 1 0 1 ) as our root element in SL(d, q), and both a short and long root element in Sp(2n, q).
We take, as our base cases, groups H of minimum rank, subject to containing root elements of all types. For example, H = SL(2, q) in the case of SL(d, q). These groups lie naturally in the larger groups of the same type, and contain our chosen root elements.
Our presentation {S | R} for C always contain four sets of relations: a presentation for N ; presentations for the subgroups H; relations that give the centraliser in N of certain root elements; and certain Steinberg relations. In some cases extra relations are required when C has an exceptional Schur multiplier.
For Chevalley groups, the sufficiency of the Steinberg relations is due to [25,27]; for the twisted groups it follows from [4]. The Steinberg relations consist of relations that hold in the root subgroups, and relations that express commutators of root elements in terms of other root elements. A sufficient number of relations of this second kind is included in R by ensuring that one is chosen, or is implied, from each orbit of the action of N on the set of unordered pairs of distinct root elements.
1.4. Comments on length. Earlier we commented that a presentation which has given bit-length N can be converted into one of word length 4N . This can be done readily as follows. Formally, we define a presentation of a group G to be of the form {X, Y | W, R}, where X is the given generating set, Y = {y 1 , . . . , y n } are auxiliary generators, W = {w 1 , . . . , w n } are words where w i is a word on X ∪ {w 1 , . . . , w i−1 } which defines y i , and R is a defining set of relators for G, expressed as words in X ∪ Y , and hence, with the use of W , as words in X. The sets Y and W are simply tools for expressing the relators R efficiently as words in X, and should not be counted either as generators or relations. For example, the generators a = ∆ x and b = ∆ y in the presentation for SU (3, q) given in Section 4 are superfluous, so we do not count a and b as generators, nor their definitions as relations. We also store words as straight-line programs on X ∪ Y ; for a formal definition and discussion of their significance, see [24, p. 10]. The most important use of straightline programs rather than words is when we raise a word to a high power. There are fast algorithms for raising a matrix over a finite field to a high power (see [19,Section 10]); so our straight-line programs encode a cell as a power of an earlier cell. Hence, for our purposes, the critical metric to assess presentation length is bit-length. 1.5. Notation. The dihedral group of order 2n, where n is positive, is The generalised quaternion group of order 2n, where n is positive and even, is The semidihedral group of order 2n, where n is positive and n ≡ 0 mod 4, is Note that we generalise the usual definitions of these groups.

Presentations for the symmetric group and related groups
Let d be a positive integer. Define ǫ d = (−1) d+1 . If ǫ = ±1, and (a 1 , a 2 , . . . , a d ) is a cyclic permutation, then we define the signed permutation (a 1 , a 2 , . . . , a d ) ǫ by a i → a i+1 for i < d, and a d → ǫa 1 . In particular, (a) − is the transposition that interchanges a and −a.
Moore [22, p. 358] gave the following presentation for S d .
Theorem 2.1. The symmetric group S d , for d > 2, has the following presentation: This presentation for S d on its standard generators has word length O(d 2 ) and bitlength O(d log d), and can easily be rewritten to have word length O(d) by introducing At the cost of introducing relations with bit-length O(d log d), we could convert the presentation for S d in [5, Theorem 1.2] of bit-length O(log d) to one on generators U and V that has an absolutely bounded number of relations.
We now give presentations for the group of signed permutation matrices of determinant 1 and degree d. By introducing auxiliary generators, these also achieve word length O(d).
Theorem 2.2. The group of signed permutation matrices of determinant 1 and degree d, when d > 2 is odd, has the following presentation: Proof. These relations are clearly satisfied. The relations [U ′ , U ′V ′j ] = 1 for 2 ≤ j ≤ (d − 1)/2 extend, by conjugation with V ′d−j , to allow 2 ≤ j ≤ d − 2. Let G denote the presented group, and A the normal closure in G of a := U ′ 2 . By the previous theorem, G/A is isomorphic to S d , the relation [U, U V j ] = 1 being equivalent to the relation (U U V j ) 2 = 1 in the presence of the relation U 2 = 1.
We prove that B := {a j+1 := a V ′j | 0 ≤ j < d} is an elementary abelian 2-group. This generating set is permuted cyclically by V ′ . The relation U ′ 2V ′ U ′ U ′ 2 U ′ 2V ′ = 1 states that a U ′ 2 a 1 a 2 = 1, so a 1 a 2 is an involution, and hence a 1 commutes with a 2 . It follows, by conjugation with V ′−1 , that a d commutes with a 1 . But a 1 commutes with a j for 3 ≤ j ≤ d − 1 since U ′ commutes with U ′V j for 2 ≤ j ≤ d − 2. So a 1 commutes with a j for all j, and conjugating by V ′ i then proves that a i and a j commute for all i and j.
We now prove that a 1 a 2 . . .
is a normal subgroup of G, and the result follows.
The following is proved similarly.
Theorem 2.3. The group of signed permutation matrices of determinant 1 and degree d, when d > 2 is even, has the following presentation: If d = 2, then the group is cyclic of order 4. We shall frequently need the following result.
Theorem 2.4. Let d > 3. The group H generated by U ′V ′2 and V ′ U ′−1 U ′−V ′ is the group of signed permutation matrices of 1, . . . , d having determinant 1 that fixes both 1 and 2. The group G generated by U ′V ′2 and V ′ U ′ U ′V ′ is the direct product of H with the cyclic group of order 2 generated by (1) − (2) − .
The subgroup of G generated by (3,4) − and its conjugates is isomorphic to H, so the extension splits.

3.
A presentation for SL(2, q) 3.1. Generators and notation. Let q = p e for a prime p. Let ω be a primitive element of GF(q). We define the following elements of SL(2, q).

3.2.
Presentations for SL(2, q) for e > 1. We use presentations from [7] rewritten on the generating set {τ = τ (1), δ, U }. A complication arises from the fact that τ (α) δ = τ (αω 2 ), so τ (ω i ) is treated differently according to the parity of i. In the presented groups given below, τ is a generator, and τ 1 , standing for τ (ω), is defined as a product of conjugates of τ by powers of δ. Now τ i may be defined for 0 ≤ i ≤ e by τ 0 = τ and Let q = p e for an odd prime p and e > 1. Let f (t) be the minimal polynomial of ω over GF(p), and let ω = e−1 i=0 a i ω 2i . Let ω 2m = 1 + ω if 1 + ω ∈ GF(q) 2 , and ω 2m+1 = 1 + ω otherwise. Then PSL(2, q) has the following presentation: This is essentially due to Campbell, Robertson, and Williams [7, Theorem 2.2]. We have replaced their generators w, x, y, z by U τ −1 , τ, τ 1 , and τ δ −1 . The proof that the elements τ i in the presented group G commute goes through unchanged with our presentation. They have relations equivalent to τ p = 1 and τ p 1 = 1. The second of these is redundant, as τ 1 is a product of commuting conjugates of τ . The relation τ f (t) = 1 gives the action of δ on the irreducible module generated by the τ i ; their equivalent relation with τ replaced by τ 1 is also redundant. Their relation z (q−1)/2 = 1 translates to (τ δ −1 ) (q−1)/2 = 1 in our notation, and we have replaced it by the relation δ (q−1)/2 = 1. Given the action of δ on the module generated by the τ i , these relations are equivalent. Finally we have replaced their relation (wyz) 3 = 1, which translates to (U τ −1 τ 1 τ δ −1 ) 3 = 1, by the relation (τ 1 U δ) 3 = 1. These are equivalent, since both [τ, τ 1 ] = 1 and U δ = δ −1 U hold in G.
Theorem 3.2. A presentation for SL (2, q), where q = p e is odd and e > 1, is obtained from the presentation for PSL(2, q) in Theorem 3.1 by replacing the relations Proof. The given relations all hold in SL (2, q). It remains to prove that U 2 is central in the presented group. The relations (τ U −1 ) 3 = U 2 and (U δ) 2 = U 2 imply that τ and δ commute with U 2 , and the result follows.
Note that τ δ (q+1)/4 is τ (−ω), and plays the role of τ 1 = τ (ω) in Theorem 3.1. It is because we can write this element as a conjugate of τ that we have a simpler presentation when q ≡ 3 mod 4.
Theorem 3.4. If q ≡ 3 mod 4 and e > 1, then a presentation for SL(2, q) is obtained from the presentation for PSL(2, q) in Theorem 3.3 by replacing the relations The proof is similar to that of Theorem 3.2.
Similar results hold for characteristic 2. Translated into our generating system, [7, Theorem 3.2] states the following.
3.4. Standard generators for SL(2, q). The standard generators for SL(2, q) are the same as the presentation generators, but with δ replaced with δ −1 .
We work with respect to the basis (e 1 , v, f 1 ) for the underlying vector space, where the hermitian form is defined by e 1 .f 1 = f 1 .e 1 = v.v = 1, and the form vanishes on all other pairs of basis vectors. Let K be the root group of order q 3 consisting of all matrices of the form where α and β lie in GF(q 2 ), and β has trace −α q+1 , for example, β = ξα q+1 . Let υ = υ(1, ξ), and let τ = υ(0, ζ) if q is odd, and τ = υ(0, 1) if q is even. Define and let ∆ = ∆(ω).
Here, for completeness, we provide a variation of their presentation on standard generators {υ, τ, ∆, t}. Let H = υ, τ, ∆ , a Borel subgroup of order (q 2 − 1)q 3 . If q is even, then relation R 1 (iii) given below defines τ , otherwise our presentation is independent of the choice of a non-trivial element of τ H ; so this difference between the presentations is insignificant. Following the existing approach, we first construct a presentation for H, and then identify explicitly the additional relations implied by [16] to obtain one for SU(3, q).

A presentation for H.
Lemma 4.1. If q = 2, 3, 5 then there exist integers x and y such that ] is equal to GF(q 2 ) or GF(q), according as p is odd or even.
Proof. This is [13,Lemma 4.23], with a and b replaced by ω −x and ω −y respectively, as can be seen by raising (i) to the power q.
The proof of this lemma in [13] allows us to obtain x and y readily. If q is even, then ω −x = ω 0 and ω −y = 1 − ω 0 , so we can solve directly for x and y. Otherwise, define ψ := ω (q+1)/2 and c := t(t 2 + 3ψ 2 )(t + ψ)(t 2 − ψ 2 ) −2 for t ∈ GF(q 2 ). If x and y satisfy conditions (i) and (ii), then ω −x = c and ω −y = c − c q−1 for some t ∈ GF(q 2 ). The number of such t in GF(q 2 ) is always positive and is at least q − 5 − 6(e − 1)q 1/2 . Hence it is easy to determine t, and so x and y, by a random search.
We repeatedly use the following which is [13,Lemma 4.5].
Lemma 4.2. Let U 0 and W 0 be subgroups of a group G, and let u, w, a, b be elements of G satisfying the following conditions: We build up our presentation for H on the generating set {υ, τ, ∆} in steps. Let G be the presented group, so that G changes as relations are imposed.
Let a = ∆ x and b = ∆ y , with x and y as in Lemma 4.1. When e > 1, we choose x such that gcd(x, q 2 − 1) = 1 whenever possible. This guarantees conditions (iii) and (iv) of the lemma.
Let U be the normal closure in H of υ, τ , so U is the group of upper unitriangular matrices in H. Let W be the normal closure of τ in H, so W is the centre of U .
We use three sets of relations to present H. The first R 1 is the following.
Suppose that m(s) = i a i s i is a polynomial over GF(p). If w ∈ W and h ∈ ∆ , we define w m(h) to be i (w a i ) h i , and if u ∈ U we define u m(h) similarly, but here the order in which the terms are multiplied is relevant. Any fixed order, such as the natural order, may be used.
We now introduce a set R 2 of relations which, as we shall see, determines, with R 1 , the structure of W = {τ δ : δ ∈ ∆ } .
To see that R 1 (ii), R 2 (i), and this last observation imply that X := {τ c : c ∈ a, b } is abelian, apply Lemma 4.2, with U 0 = W 0 = 1 , and u = w = τ . Since a and b commute, R 2 (i) implies that X = {τ c : c ∈ a } . If e = 1 then (iii) asserts that ∆ normalises τ , so this group is equal to W , as required. If e > 1 then (ii) implies that X has order p e . If also gcd(x, q 2 − 1) = 1 then ∆ ∈ a , so ∆ normalises X, which is hence equal to W , as required. But if gcd(x, q 2 − 1) > 1 then (iii) implies that X is normalised by ∆, as required.
We now introduce a set R 3 of relations that imply, with R 1 and R 2 , that U/W is elementary abelian of order q 2 .
Observe that (i) is satisfied in the matrix group since ω x(q−2) + ω y(q−2) = 1. The elements w i need to be determined by calculation in the matrix group. We first show that these relations imply that U/W 1 is elementary abelian, of order q 2 , where W 1 = {τ g : g ∈ G} . Let Y = { υ, τ c : c ∈ a, b } < G, and Z = Y W 1 /W 1 . Suppose first that p is odd. The fact that Z is abelian follows from Lemma 4.2 with u = w = υ, and U 0 = W 0 = W 1 . Since a and b commute, it follows from (i) that Z = {υ c : c ∈ a } W 1 /W 1 . The fact that Z is elementary abelian follows from R 1 (iii). It now follows from (v) that Z has order at most q 2 , and hence has order exactly q 2 . Since ∆ commutes with a, it follows from (vi) that Y is normalised by ∆. So Y = U , and U/W 1 has order q 2 , as required. If p is even, then we need to prove that Z has order q, and that U/W 1 is the direct sum of Z and Z ∆ . The proof that Z is elementary abelian goes through as in the case when p is odd. The fact that Z has order q follows from (v). The fact that Z and Z ∆ commute follows from Lemma 4.2 with u = υ, and w = υ ∆ , and U 0 = W 0 = W 1 . The fact that the direct sum of these two groups is normalised by ∆ follows from (vi). It remains to prove that W 1 = W ; in other words, that W is normalised by U . Since W is normalised by ∆, it suffices to prove that it is normalised by υ, or, equivalently, by υ a . We have seen that W = {τ c : c ∈ a, b } . But [ {τ c : c ∈ a, b } , {υ c : c ∈ a, b } ] = 1 by Lemma 4.2, with u = τ , and w = υ, and U 0 = 1 , and W 0 = W .
It follows that G has order at most q 3 (q 2 − 1), and hence has precisely this order. Our analysis implies the following.

4.2.2.
Completing the presentation for SU (3, q). We follow [16] with some variations. Our presentation for G is where R(1) are the relations in a presentation for H on {υ, τ, ∆}, and R(2) is to be specified. We omit the relation ∆ q 2 −1 = 1 from R(1): it is an obvious consequence of ∆ t = ∆ −q and t 2 = 1. Since our defining relations for H are satisfied in SU(3, q), we may take H to be a subgroup of G. The relations R(2) are of the form u t = u L dtu R for u, u L , u R ∈ U , and d in D := ∆ ; in more detail, if u = υ(α, β) and β = 0, so u = 1, then d = diag(β −q , β q−1 , β), u L = υ(−αβ −q , β −1 ), and u R = υ(−αβ −1 , β −1 ). Note that SU(3, q) is the disjoint union of U DtU and H. Thus U DtU is the set of elements that move the isotropic projective point [0, 0, 1], and H is the set of elements that fix [0, 0, 1]. Hence it suffices to find relations R(2) that imply that every element of G lies in U DtU ∪ H, where we abuse notation by using the same letter for t as an element of G, and as a matrix. We denote the relation u t = u L dtu R by P (u).
Let V denote the set of u ∈ U \ 1 G for which the relation P (u) may be deduced from the above presentation for a given choice of R(2).
Proof. It is easy to see from these relations that the product of two elements of G of the form u L dtu R can again be written in this form, or as an element of H. Thus the union of these subsets of G is a subgroup that contains the generators of G. Hence G is equal to this union, and so is isomorphic to SU(3, q). Proof. Observe that (u ∆ ) t = (u t ) (∆ t ) , and ∆ t = ∆ −q . Also t ∆ = (∆ −1 ∆ t )t. The result follows. Proof. This is simply a rearrangement of terms.
It suffices to prove that this choice of R(2) implies that V = U \ {1}.
is the disjoint union of the U i , as i runs from 0 to q. Moreover U 0 contains q − 1 elements, and U i contains Each U i is normalised by ∆ , the action being transitive in all cases.
Proof. Note that ω q−1 is an element of GF(q 2 ) × that has norm 1 and is not a cube. If α q+1 = −tr(β 0 ) then α and αω q−1 and αω 2(q−1) all have norm equal to −tr(β 0 ), and lie in different cosets of GF(q 2 ) × 3 in GF(q 2 ) × ; so they may be taken to be the elements α i of the theorem.
Define U i , where 0 ≤ i ≤ q, as in the proof of Theorem 4.8. Again ∆ acts transitively on U 0 , but partitions U i , for 1 ≤ 1 ≤ q, into three orbits, the orbit to which a given element υ(α, β) of U i belongs depending on the coset of GF(q 2 ) × 3 in GF(q 2 ) × to which α belongs.
The proof now follows that of Theorem 4.8.

The remaining cases.
A presentation for SU(3, 3) is the following: A presentation for SU (3,5) is the following: Their correctness is readily established using coset enumeration.

5.
A presentation for SL(d, q) for d > 2 5.1. Generators and notation. Let q = p e for a prime p. Let ω be a primitive element of GF(q). Let d > 2. We take a basis (e 1 , e 2 , . . . , e d ) of the natural module. We define the following elements of GL(d, q), where 1 ≤ i, j ≤ d and i = j, and s ∈ GF(q).
All these elements of GL(d, q) centralise those basis elements that they are not stated to move. All lie in SL(d, q) except for U when q is odd, and V when q is odd and d is even. If q is even, then U ′ = U and V ′ = V .
Let N be the subgroup of SL(d, q) generated by {δ, U ′ , V ′ }. So N is an extension of the direct product of d − 1 copies of C q−1 , generated by {δ 12 , δ 23 , . . . , δ d−1,d }, by a copy of S d , and has order (q − 1) d−1 d!.
Theorem 5.1. Let G be the group generated by {δ, U ′ , V ′ } subject to the relations given below. Then G is isomorphic to N .
Proof. Since q ≥ 4, the elements δ ij are distinct. The centraliser of δ in U ′ , V ′ is generated by the subgroup that, as signed permutation group, fixes 1 and 2, and by U ′ 2 and U ′ 2V ′ if q is odd. For these elements centralise δ, and generate a subgroup of U ′ , V ′ of the appropriate index. But U ′ 2 ∈ δ , and U ′ 2V ′ ∈ δ V ′ , so, by (ii) and (v), we may define δ ij in G, these elements being permuted by the natural action of U ′ , V ′ acting as unsigned permutations. By (v) δ 12 commutes with δ 23 , and with δ 34 if d > 3; so all δ ij commute with each other. By (iv) in N and is generated by the following elements: Theorem 5.3. Let q = p e for a prime p and e > 1, and let d > 2. Let G be the group generated by {τ, δ, U ′ , V ′ } subject to the relations given below. Then G is isomorphic to SL(d, q).
(i) Defining relations for N = δ, U ′ , V ′ as in Theorem 5.1, but omitting relations (iv) and (vi) of that theorem.
Relations that state that the elements listed in Theorem 5.2 centralise τ . (iv) The following instances of Steinberg relations: Proof. The omitted relations follow from (ii).
By (iii) the elements τ ij (s) (for s = 0) may be uniquely defined in G as N -conjugates of τ .
The Steinberg relations for SL(d, q) are as follows, where i, j, k and l are distinct: Relations (1) hold in G for i = 1 and j = 2 from (ii), and for other suffices by conjugation in N .
In SL(d, q), if d > 4, or if d = 4 and q is even, then N acts transitively by conjugation on the set of ordered pairs (τ ij (s), τ kl (t)), where i, j, k, l are distinct and s = 0 = t. So in our presentation, for these values of d and q, we need one relation of the form [τ ij (s), τ kl (t)] = 1 with suffices satisfying these conditions. Relation (d) is one such. Thus relations (2) hold in G if d > 4, or if q is even. If d = 4 and q is odd then the action of N on these pairs gives rise to two orbits, one containing the pairs (τ ij (s), τ kl (t)) where st is a square in GF(q) × , and the other containing the pairs where st is not a square. Now for fixed i, j, k, l, t, the set of s ∈ GF(q) for which [τ ij (s), τ kl (t)] = 1 in G is closed under addition (by (a)), and under multiplication by ω 2 (by conjugation with δ ij ). Since ω 2 generates GF(q) as a field, [τ ij (s), τ kl (t)] = 1 for all s in GF(q) provided that this holds for one s ∈ GF(q) × , and so (2) holds in G.
Similarly, if d > 3 then N acts transitively by conjugation on the set of pairs (τ ij (s), τ ik (t)), where i, j, k are distinct and s = 0 = t; and we have the relation (b), standing for [τ 12 (1), τ 13 (1)] = 1. But if d = 3 then the setwise stabiliser of the set of pairs (τ 12 (s), τ 13 (t)), with s = 0 = t, is the group of diagonal matrices in SL(d, q), and this group, which has order (q − 1) 2 , acts faithfully unless q ≡ 1 mod 3, when there is a kernel of order 3. So if q ≡ 1 mod 3 then we have three orbits of N on these pairs. For fixed s ∈ GF(q) × , let S be the set of t ∈ GF(q) such that [τ 12 (s), τ 13 (t)] = 1 in G. Now S is closed under addition, and (by conjugation with δ 12 δ 2 23 ) is closed under multiplication by ω 3 , and hence is a vector space over the field generated by ω 3 . If q = 4 then this field is GF(q), and a similar result holds with the roles of s and t interchanged. Thus, in this case, the relations (3) may be deduced from a single case. But if q = 4 the field generated by ω 3 is the prime field, and we need relation (f), which asserts that [τ 12 (1), τ 13 (ω 2 )] = 1. Hence [τ 12 (1), τ 13 (1 + ω 2 )] = 1, so we have representatives of the three orbits of N , as above, thus (3) holds in all cases.
By a similar argument, relations (4) may be deduced, using relations (c) and (g). The proof of (5) is similar. If q ≡ 1 mod 3, or d > 3, then the result follows from (a) by conjugation with N . Otherwise, we first need to prove that in G, for given s ∈ GF(q) × , the set S consisting of those t for which [τ 12 (s), τ 23 (t)] = τ 13 (st) is closed under addition, and under multiplication by ω 3 . This follows from the identity [a, bc] = [a, c][a, b] c , using the relation (a), and by conjugation with δ 12 δ 2 23 . This completes the proof of (5), except in the case when q = 4 and d = 3, when we use relation (e).
Finally one must check that the given generators lie in the group generated by the τ ij (s, t). This holds for τ , δ and U ′ , thus for all N -conjugates of U ′ , and so for V ′ .
Coset enumeration shows that relations (f) and (g) are redundant.

5.3.
A presentation for SL(d, p). We give a presentation for SL(d, p) on the generating set {τ, U ′ , V ′ }. Clearly U ′ , V ′ acts transitively by conjugation on the set of elements τ ij (±1), a set of size 2d(d − 1) if p is odd, and of size d(d − 1) if p = 2. If p is odd, then the centraliser of τ in U ′ , V ′ is the direct product of the cyclic group of order 2 generated by U ′2 with the subgroup of U ′ , V ′ that, as signed permutation group, fixes 1 and 2. This subgroup has index 2d(d − 1), as required. If p = 2, then this centraliser is the subgroup of U, V that fixes 1 and 2, and has index d(d − 1), as required.
Theorem 5.4. Let p be a prime and let d > 2. Let G be the group generated by {τ, U ′ , V ′ } subject to the relations given below. Then G is isomorphic to SL(d, p).
Proof. That U ′2 centralises τ follows from (ii). Hence, by virtue of (i) and (iii), if p is odd then we may define τ ij (±1) as elements of G by conjugation with U ′ , V ′ . The proof is concluded as in Theorem 5.3.
is a diagonal matrix whose entries, with one possible exception, equal −ǫ d ω.

5.5.
Standard generators for SL(d, q). The inverse of the presentation generator δ is the standard generator δ; the standard generator if q is even.

6.
A presentation for Sp(2n, q) for n > 1 6.1. Generators and notation. Let q = p e for a prime p. Let ω be a primitive element of GF(q). Let n > 1. We take a hyperbolic basis (e 1 , f 1 , . . . , e n , f n ) of the natural module, and define the following elements of Sp(2n, q), where 1 ≤ i, j ≤ n and i = j, and s ∈ GF(q).
All these elements of Sp(2n, q) centralise those basis elements that they are not stated to move. Note that

6.2.
A presentation for Sp(2n, q) and e > 1. We give a presentation for Sp(2n, q) on the generating set If q is even, then the subgroup of Sp(2n, q) obtained by omitting τ is SO + (2n, q).
It is isomorphic to C 2 ≀ S n if q is even and to C 4 ≀ S n if q is odd. The following is clear.
The centraliser of σ = σ 12 (1) in N is generated by the following elements: The centraliser of τ = τ 1 (1) in N is generated by the following elements: Proof. Observe that N acts transitively by conjugation on the set of 2n(n − 1)(q − 1) short root elements σ ij (s), where 0 < |i| = |j| ≤ n and s ∈ GF(q) × , bearing in mind that σ ij (s) = σ −j,−i (−s) for any i, j, s. The centraliser of σ in N contains the direct product of a subgroup that is isomorphic to K ≀ S n−2 with another copy of K, namely δ 1 δ 2 , ZU Z −1 . Since this direct product has index 2n(n − 1)(q − 1) in N , it is the centraliser of σ. Also, N acts by conjugation on the set of 2n(q − 1) long root elements τ i (s), where 0 < |i| ≤ n and s ∈ GF(q) × . The action is transitive if q is even, and there are two orbits of equal size if q is odd, τ i (s) falling into one or other orbit according as s is or is not a square. The elements U and V permute the suffices, positive or negative, of the root elements τ i (s) in the natural way, and Z i interchanges the suffices i and −i and fixes all other suffices.
Thus the stated elements centralise τ , and generate a subgroup of N of index 2n(q − 1) if q is even, and n(q − 1) if q is odd.
Proof. The isomorphism is obtained by restriction to the subspace e 1 , e 2 , or f 1 , f 2 , these two representations being connected by the inverse transpose automorphism.
Theorem 6.5. Let q = p e for a prime p and e > 1, and let n ≥ 2. Let G be the group generated by {σ, τ, δ, Z, U, V }, omitting V if n = 2, subject to the relations given below. Then G is isomorphic to Sp(2n, q).
(i) Defining relations for N = δ, Z, U, V , omitting V if n = 2, as in Theorem 6.2, but omitting relations (ii) of that theorem, and relation (ii) of Theorem 6.1. (ii) Relations that state that the elements listed in Theorem 6.3 centralise σ and τ , but omitting the relation [τ, The given relations are instances of the more general relations in parentheses, with s = t = 1 and (i, j, k, l) = (1, 2, 3, 4). In the parenthetical version, the moduli of i, j, k, l are distinct.
Proof. The double suffices (i, j) for σ and the suffices i for τ correspond to short and long roots respectively; and if r is the root corresponding to (i, j) or i, then −r is the root corresponding to (−i, −j) (equivalently to (j, i)) or −i respectively.
The relations omitted from (i) follow from (iv). By (i) we may define the elements {δ i , Z i , U, V } in G, and hence also the elements ∆ and U ′ . By Theorem 6.3, and using (i) and (ii), we may define {σ ij (s) | 0 < |i| = |j| ≤ n, s ∈ GF(q) × }, as a subset of G.
The relation [τ, V U ] = 1, omitted from (ii) if q is odd, may now be supplied as follows. It follows from (v)(e) that the subgroup of U, V that centralises 1 and 2, as permutations of {1, 2, . . . , n}, centralises τ as a subgroup of G. But this subgroup, together with U V (see (ii)), generates the symmetric group on {2, 3, . . . , n}, and hence contains V U . By Theorem 6.3, and using (i) and (ii), we may define {τ i (s) | 0 < |i| ≤ n, s ∈ GF(q) 2 } as a subset of G, and obtain the centralisers of these elements in N . Suppose now that q is odd. Using (iv) we may define τ 1 (ω) in G, and writing ω as a sum of squares gives τ 1 (ω) as a product of elements of the form τ i (t 2 ) for t ∈ GF(q) × . But these elements all have the same centraliser C in N . So C also centralises τ 1 (ω); since it has the expected index in N it must be the full centraliser. So we may define τ i (s) for all i, and for all s ∈ GF(q) \ GF(q) 2 , as N -conjugates of τ 1 (ω).
We now turn to the Steinberg relations. The relations σ ij (s + t) = σ ij (s)σ ij (t) and σ ij (s) p = 1 follow from (ii) and (iii), and the relations τ i (s + t) = τ i (s)τ i (t) and τ i (s) p = 1 follow similarly from (ii) and (iv).
The remaining Steinberg relations are (a) to (j), and we prove that the general case, given in parentheses, is implied by the given relation.
The general case of relations (a) to (d) clearly follow from the one given instance by conjugation with N .
For fixed i, j, and s the set of values t for which the general case of relation (f) holds is closed under addition and multiplication by ω 2 (conjugate by δ i δ j ) and hence is either GF(q) or {0}, since ω 2 generates GF(q) as a field. It follows, from conjugation by N , that all instances of (f) follow from the given instance.
Similarly the general cases of (h) and (j) follow from a single case.
To prove the general case of (g), note that τ j (t) commutes with the RHS of this identity by (f) and (j), since σ i,−j (st) = σ j,−i (−st). The proof is similar to that of (e).
Variants of (e), (f), and (g) may be obtained by changing signs of suffices, but these follow from the given relations. For example, for i and j positive, So all the Steinberg relations hold in G. Thus the subgroup H of G generated by the Steinberg generators is isomorphic to Sp(2n, q). Also, Z and δ and U ′ lie in this group by (iii) and (iv), and hence U lies in H. But H is normalised by U, V , and U, V is isomorphic to S n , by (i), so V ∈ H. Thus G = H, as required. Theorem 6.6. The centraliser of σ = σ 12 (1) in N 1 is generated by the corresponding elements (i) − (iv) listed in Theorem 6.3, and [Z 2 , U ] if p is odd. It is isomorphic to The centraliser of τ = τ 1 (1) in N 1 is generated by the corresponding elements (i) − (iv) listed in Theorem 6.3. It is isomorphic to C 2 ≀ S n−1 if p = 2, and to Proof. The listed elements clearly generate groups having the claimed isomorphism types. They generate subgroups of N 1 of index 2n in the case of the centraliser of τ ; and of index 4n(n − 1) if p is odd, and index 2n(n − 1) if p = 2, in the case of the centraliser of σ. This corresponds to the fact that N 1 permutes transitively the 2n elements τ ±i (1) with 1 ≤ i ≤ n, and the 4n(n − 1) elements σ ±i,±j (±1) with 1 ≤ i = j ≤ n if p is odd, and the 2n(n − 1) elements σ ±i,±j (1) with 1 ≤ i = j ≤ n if p = 2. (Recall that, in either characteristic, these elements are equal in pairs.) Theorem 6.7. Let p be a prime and let n ≥ 2. Let G be the group generated by {σ, τ, Z, U, V }, omitting V if n = 2, subject to the relations given below. Then G is isomorphic to Sp(2n, p). Proof. The proof is similar to that of Theorem 6.5.
6.4. A presentation for PSp(2n, q). If q is even, then Sp(2n, q) is simple. If q is odd, then the centre of Sp(2n, q) has order 2, and is generated by (ZV ) 2n .
6.5. Standard generators for Sp(2n, q). The inverse of the presentation generator δ is the standard generator δ. The standard generator x = σ Z .

7.
A presentation for SU(2n, q) for n > 1 7.1. Generators and notation. Let q = p e for a prime p. Let ω be a primitive element of GF(q 2 ), and let ω 0 = ω q+1 , so ω 0 is a primitive element of GF(q). Define ψ = ω (q+1)/2 if q is odd, and ψ = 1 if q is even. Let n > 1. We take a hyperbolic basis (e 1 , f 1 , . . . , e n , f n ) of the natural module. We define the following elements of SU(2n, q), where 1 ≤ i, j ≤ n and i = j, and s ∈ GF(q), and α ∈ GF(q 2 ).
All these elements of SU(2n, q) centralise those basis elements that they are not stated to move. As in the case of Sp(2n, q), σ ij (α) Note that the change of basis that fixes e i and sends f i to −ψ −1 f i for all i has the effect of replacing ψ by 1 in these definitions. Thus the isomorphism type of the group generated by these elements is independent of ψ. So the value of ψ plays no role in the presentation and proofs, apart from the fact that ψ = 0. We choose ψ to ensure that these elements preserve the given form.
Let H be the subgroup of SU(2n, q) generated by {∆, U, V }, omitting V if n = 2. Clearly H has order (q 2 − 1) n−1 n!. Proof. The result is clear if n = 2, so assume n > 2. Relations (ii) state that ∆ is centralised by the subgroup of U, V that corresponds to the copy of S n−2 that fixes 1 and 2, so the conjugates of ∆ under U, V may be labelled as ∆ ij , where 1 ≤ i = j ≤ n. Relation (iii) states that ∆ ji = ∆ −1 ij when (i, j) = (1, 2), and hence for any suffices (i, j). Relation (v) states that ∆ 12 commutes with ∆ 23 and ∆ 34 , when these are defined, and hence all ∆ ij commute with each other. Finally relation (iv) states that ∆ i,i+1 ∆ i+1,i+2 = ∆ i,i+2 when i = 1, and hence for all i. Hence, in G, every element of the group generated by the ∆ ij may be written in the form ∆ m 1 12 ∆ m 2 23 · · · ∆ m n−1 n−1,n , and the theorem is proved. Let K be the subgroup of SU(2n, q) generated by {δ, ∆, U, V }, omitting V if n = 2. Clearly K has order (q − 1)(q 2 − 1) n−1 n!. Proof. Suppose first that n > 2. Relations (iii) state that δ is centralised by the subgroup of U, V that corresponds to the subgroup of S n−1 that fixes 1, so the conjugates of δ under U, V may be labelled as δ i , where 1 ≤ i ≤ n. Relation (v) states that δ 2 = δ 1 ∆ −(q+1) ; so, using (iv), δ 1 commutes with δ 2 , and hence δ i commutes with δ j for all i, j. By (iv), δ commutes with ∆ 12 and with ∆ 23 , and hence δ i commutes with ∆ jk for all i and for all j = k. It follows from (v)

then
A is an abelian group, and A/B is generated by {δ i | i > 0} subject to the relations δ i = δ j , and δ q−1 1 = 1. So the group presented has order at most (q − 1)(q 2 − 1) n−1 n!. But the corresponding matrix group has this order, so the theorem is proved when n > 2. The case n = 2 is proved similarly.
Let N be the subgroup of SU(2n, q) generated by {Z, δ, ∆, U, V }, omitting V if n = 2. It has order (q − 1)(q 2 − 1) n−1 2 n n!. Proof. Suppose first that n > 2. As with the previous theorem, relations (iii) imply that we have elements Z i in G that are permuted by U, V via the natural action of S n on the suffices. Relation (iv) asserts that [Z 1 , Z 2 ] = 1; so the group generated by the Z i is abelian. Relation (vi) states that ∆ 23 is centralised by for all j = i, and hence δ i is centralised by Z k for all k = i since n > 2. It now follows, from (vii), that the group A in the proof of the previous theorem is normalised by Z 1 , and hence by Z i for all i. By (ii), A contains Z 2 1 , and hence contains Z 2 i for all i. Thus G has at most the stated order; since the corresponding matrix group has this order the theorem is proved for n > 2.
The proof for n = 2 is similar, except for the proof that δ i commutes with Z j for all i = j. This follows from (viii), which states that δ 1 commutes with Z 2 if n = 2.
Proof. The first statement is obvious. Observe that H acts on e 1 , e 2 as SL(2, q 2 ), and similarly on f 1 , f 2 , the actions corresponding under the inverse transpose automorphism followed by the Frobenius automorphism.
Theorem 7.5. The centraliser H of σ = σ 12 (1) in N is generated by the following elements: and q is odd, and δ V 2 if n = 3 and q is odd; But the 4n(n−1)(q 2 −1) matrices defined are equal in pairs, as shown in Section 7.1. The orbit has size 2n(n − 1)(q 2 − 1), so H has order (q − 1)(q 2 − 1) n−2 2 n−1 (n − 2)!. Let K be the group generated by the itemised elements. Clearly K is a subgroup of H. The subgroup K 0 of K generated by the elements in (i) is the copy of S n−2 that permutes the basis vectors {e 1 , e 2 , . . . , e n }, fixing e 1 and e 2 , and also permutes {f 1 , . . . , f n }, fixing f 1 and f 2 .
Finally let n = 2. The orbit of σ under conjugation by N is {σ ±1,±2 (α 2 ) | α ∈ GF(q 2 ) × }, and hence has size 4(q 2 − 1) in N if q is even, and 2(q 2 − 1) if q is odd. So the centraliser of σ has order 2(q − 1) if q is even, and order 4(q − 1) if q is odd. Items (v) and (vi) generate a copy of D 2(q−1) if q is even, else a copy of SD 4(q−1) ; and these groups have the required order.
Theorem 7.6. The centraliser of τ = τ 1 (1) in N has order (q 2 − 1) n−1 2 n−1 (n − 1)! and is generated by the following elements: (ii) if n = 2 and q is odd then δ U , and if n > 2 then ∆ V ; Proof. Recall that the order of N is (q−1)(q 2 −1) n−1 2 n n!. The orbit of τ under conjugation by N is {τ ±i (s) | 1 ≤ i ≤ n, s ∈ GF(q) × }, as τ i (s) ∆ = τ i (sω 0 ), and has size 2(q − 1)n. The elements in (i) imply that τ is centralised by the copy of S n−1 in U, V that fixes the basis vectors e 1 and f 1 .
If n > 2 and q is even, then the proof is simpler: ∆ 2 δ −1 and the S n−1 -conjugates of ∆ V now generate a group of order (q 2 − 1) n−1 .
Theorem 7.7. Let q = p e for a prime p and let n ≥ 2. Let G be the group generated by {σ, τ, Z, δ, ∆, U, V }, omitting V if n = 2, subject to the relations given below. Then G is isomorphic to SU(2n, q).
The relation [τ, V U ] = 1 omitted from (iii) may be supplied using relation (v)(b) exactly as with the presentation for Sp(2n, q); here the argument applies in all characteristics.
As to (v) Relations (i), (ii) and (iii), together with (iv) if n = 2, allow us to define elements σ ij (α) and τ i (s) of G that are permuted by N .
The following Steinberg relations must now be proved to hold in G. Here i, j, k, l are distinct positive integers in the range [1, n], and α, β ∈ GF(q 2 ), and s, t ∈ GF(q). Similar relations hold when we lift the restriction that i, j, k and l are positive. These relations are obtained from those listed by conjugating with Z i , Z j , Z k , and Z l , as appropriate, and may differ from them in the signs of the coefficient: σ −i,j (α) Z i = σ i,j (−α) for i > 0 and j positive or negative, and similarly σ i,−j (α) Z j = σ i,j (−α) for j > 0 and i positive or negative. It suffices to prove that these relations hold in G when the suffices i to l are positive. We assume that (i, j, k, l) = (1, 2, 3, 4); the general case follows by conjugation in U, V .
Relation (vi) is also required, since the Steinberg relations define the universal covering group of the simple group, and this is not the classical group in the case of PSU(2n, q) when (n, q) is (2, 2), or (2, 3), or (3,2). Our presentations include relations that are not Steinberg relations, in particular those for N . We find, by coset enumeration, that our presentations in these three cases are correct; in particular, no further relation is needed if (n, q) is (2, 2) or (3,2). Note that SU(4, 2) is simple, and its universal covering group is isomorphic to Sp(4, 3), which has centre of order 2.
The relations in (11) follow from (8)-(10) by conjugating with Z i , Z j and Z k , as appropriate, and using (1).
If n > 3 then relations (f) and (g) may be supplied using (h) and (b). Relations (12) now follow.
It follows that the conjugates of σ and τ under N = Z, ∆, U, V , omitting V if n = 2, generate a subgroup H of G that is isomorphic to SU(2n, q). But Z and ∆ and U lie in H, as do their conjugates under N , and hence if n > 2 then V lies in H, as a product of N -conjugates of U . Thus G = H, as required.
We shall need the following result.
Corollary 7.8. If q is odd, then a presentation for SU(4, q)/ −I 4 is obtained from the presentation for SU(4, q) of Theorem 7.7 by replacing the presentation for SL(2, q 2 ) on {σ, ∆, U Z 2 } by a presentation of PSL(2, q 2 ) on these generators as in Theorem 3.1.
Let n > 1. We take a basis (e 1 , f 1 , e 2 , f 2 , . . . , e n , f n , w) of the natural module, with symmetric bilinear form defined by e i .f i = f i .e i = 1 for all i, and w.w = 1, and the form vanishes on all other pairs of basis vectors. So SU(2n + 1, q) contains SU(2n, q) as defined in Section 7.1. We retain the notation of that section and define the following new elements of SU(2n + 1, q), where 1 ≤ i ≤ n and α ∈ GF(q 2 ).
All these elements of SU(2n + 1, q) centralise those basis elements that they are not stated to move.
Let N be the subgroup of SU(2n + 1, q) generated by {Γ, t, U, V }, omitting V if n = 2. It is isomorphic to (C q 2 −1 : C 2 ) ≀ S n , where the 2-cycle generated by t acting on the (q 2 − 1)-cycle generated by Γ acts as g → g −q .
Theorem 8.1. Let G be the group generated by {Γ, t, U, V }, omitting V if n = 2, subject to the relations given below. Then G is isomorphic to N .
Proof. This is a standard wreath product presentation.
The case n = 2 is obvious.
The centraliser of σ = σ 12 (1) in N has index 2(q 2 − 1)n(n − 1) in N and is generated by the following elements: Proof. The elements {σ ±i±j (α) | i ≤ i = j ≤ n, α ∈ GF(q 2 ) × } of SU(2n + 1, q), each of which may be written in exactly two ways in this form, are permuted transitively by N , forming an orbit of size 2(q 2 − 1)n(n − 1). Clearly the itemised elements centralise σ. The first two sets generate a copy of (C q 2 −1 : C 2 ) ≀ S n−2 , and the last two a copy of C q 2 −1 : C 2 . Their direct product has the claimed index in N .

(vi) The following instances of Steinberg relations:
; (see (5)  Proof. The relations omitted from (i) are supplied by (iv), so we may define N as a subgroup of G. The relations omitted from (iii) may be retrieved using (i), (ii), and (a).
In SU(2n + 1, q) the centraliser of τ in N is the direct product of the cyclic group of order q + 1 generated by Γ q−1 with the subgroup of N that fixes e 1 and f 1 , and it has order 2 n−1 (q + 1)(q 2 − 1) n−1 (n − 1)!. This direct product has index 2(q − 1)n in N , which corresponds to the fact that N acts transitively by conjugation on {τ ±i (s) | 1 ≤ |i| ≤ n, s ∈ GF(q) × }. It follows from (i), (ii), (iv) and (j) that this group centralises τ in G, so {τ ±i (s) | 1 ≤ |i| ≤ n, s ∈ GF(q) × } may be be defined in G, together with the action of N on these elements.
The normaliser N of our chosen torus in SU(2n + 1, q) contains the normaliser of our chosen torus in SU(2n, q) as a subgroup of index q + 1; but the centraliser of σ in N is correspondingly larger, so the definitions of the Steinberg generators σ ij (α) are essentially the same in both cases. Thus G contains a copy of SU(2n, q), and our first objective is to prove that the Steinberg relations involving the generators υ i (α), and not subsumed in (iv), are satisfied. These relations are as follows, where |i|, |j|, and |k| are distinct: We have omitted Steinberg relations whose LHS involves the same suffix twice, but with opposite signs, since σ −i,j (α) = σ −j,i (α q ).
We may always assume that (i, j, k) = (1, 2, 3) since conjugating a Steinberg generator by t i changes the sign of suffices of the form ±i and keeps all other suffices fixed.
Thus the subgroup H of G generated by the Steinberg generators is isomorphic to SU(2n + 1, q), and it remains to prove that this subgroup equals G. By (iii), Γ and t lie in H. Also, U and V lie in H, as in the case of SU(2n, q). Thus G = H, as required. PSU(2n + 1, q). The centre of SU(2n + 1, q) has order gcd(q + 1, 2n + 1); if it is non-trivial, then it is generated by (ΓZV ) 2n(q+1)/ gcd(q+1,2n+1) . SU(2n + 1, q). The presentation generator σ = [υ tU , υ t ] t and so can be obtained in terms of standard generators.
9.1.1. Generators and notation. Let q = p e for an odd prime p. Let ω be a primitive element of GF(q).
if n is even, so the statements concerning V follow. 9.1.2. A presentation for Ω + (4, q). Since Ω + (4, q) is the central product of two copies of SL(2, q), there is no difficulty in writing down the required presentation, but we look more closely at this case as a preparation for the general case.
Let N 1 be the subgroup of Ω + (2n, q) generated by {Z, U ′ , V ′ }. This is an extension of an elementary abelian 2-group A of rank 2(n−1) by a copy of S n . Here A is the direct product of a subgroup of rank n − 1, generated by U ′2 and its conjugates, with a second subgroup of the same rank generated by Z and its conjugates. So if B is the copy of the four-group generated by the transformations (e 1 → −e 1 , f 1 → −f 1 ) and (e 1 → f 1 , f 1 → e 1 ), the other basis vectors being fixed, then N 1 is the subgroup of B ≀ S n of index 4 consisting of the elements of determinant 1 and spinor norm 1.
Proof. Relations (ii) and (iii) state that Z is centralised by the subgroup of U ′ , V ′ that centralises 1 and 2 as signed permutations, and also by U ′2 . These relations, together with (v), also imply that Z is centralised by all U ′ , V ′ -conjugates of U ′2 , so we may label these conjugates as Z ij , where 1 ≤ i = j ≤ n. Now (vii) implies that Z 12 = Z 21 , and hence, by conjugation, that Z ij = Z ji for all i = j. It follows similarly from (viii) and (ix) that all Z ij commute, and hence, by (iv), generate an elementary abelian 2-group. Also, (vi) states that Z 12 Z 23 = Z 13 , so Z ij Z jk = Z ik for any three distinct values of i, j, k, and so the subgroup of G generated by the Z ij is generated by {Z 12 , Z 23 , . . . , Z n−1,n }, and hence has order at most 2 n−1 . Thus G is a split extension of a normal subgroup of order at most 2 n−1 by the group U ′ , V ′ of order 2 n−1 n!, by (i). But N 1 , as a subgroup of Ω + (2n, q), has order 2 2n−2 n!, and the result follows.
Theorem 9.2. Let q = p e for an odd prime p and e > 1, and let n ≥ 3. Let G be the group generated by {∆, Z, U ′ , V ′ } subject to the relations given below. Then G is isomorphic to N .
(i) Defining relations for N 1 = Z, U ′ , V ′ as in Theorem 9.1.
Recall that Ω + (4, q) is isomorphic to the direct product of two copies of SL(2, q) with the centres amalgamated, the natural representation of Ω + (4, q) being the tensor product of two copies of the natural representation of SL(2, q). Here σ 12 , ∆ 1,−2 , U ′ is one copy of SL(2, q), and σ 1,−2 , ∆ 12 , Z 12 U ′ is the other. The first of these subgroups in G is isomorphic to SL(2, q) by (iii), and, since Z 12 U ′ = U ′Z 23 , so is the second, by conjugation with Z 23 . By (e), (i) and (ii), these subgroups commute with each other. The amalgamation of the centres follows from the presentation for N . It follows that, in G, the subgroup σ ±1,±2 (1), ∆ 12 , ∆ 1,−2 , Z, U ′ is isomorphic to Ω + (4, q).
The parenthetical Steinberg relations under each of the headings (a) to (d) all follow from one example, by conjugation with N .
All remaining Steinberg relations may be deduced as follows. Those that only involve positive suffices follow easily, as in the case of SL(n, q). Indeed, σ, ∆, U ′ , V ′ is isomorphic to SL(n, q). Since σ ±1,±2 (1), ∆ 12 , ∆ 1,−2 , Z, U ′ is isomorphic, in G, to Ω + (4, q), those Steinberg relations that only involve two suffices, with various signs, all hold in G. Relations that involve four different suffices, with various signs, follow from the one case (d) by conjugation, except in the case n = 4, when the extra relation (f) is needed, as we can only change the signs of the suffices two at a time. It remains to consider the case when three suffices occur: namely, the LHS of the relation is of the form [σ ij (s), σ uv (t)], where {a, b, c} := {|i|, |j|, |u|, |v|} forms a set of size 3. Conjugating by Z ab , or Z bc , or Z ac changes the signs of any two of these suffices at all their appearances. Also the relation σ ij (s) = σ −j,−i (−s) allows us to change the sign of i and j (or of u and v). Hence we reduce to the case in which all suffices are positive.
Thus all Steinberg relations hold in G, and it follows that the subgroup H of G generated by the Steinberg generators is isomorphic to Ω + (2n, q). By (iii) H contains U ′ , so it contains all the N -conjugates of U ′ , and hence contains V ′ . The result follows. 9.1.4. A presentation for Ω + (2n, p). We give a presentation for Ω + (2n, p) on the generating set {σ, Z, U ′ , V ′ }. Observe that N 1 = Z, U ′ , V ′ , as a subgroup of Ω + (2n, p), acts transitively by conjugation on the set of elements σ ij (ǫ), where |i| = |j| and ǫ = ±1. For example, σ 12 (1) U ′ 2V ′ = σ 12 (−1). The number of symbols of this form is 8n(n − 1); but as group elements they are equal in pairs, since σ ij (s) = σ −j,−i (−s). The centraliser of σ in N 1 is the direct product of the cyclic group of order 4 generated by Z 12 U ′ with the subgroup of N 1 that has index 4n(n − 1) and centralises e 1 , f 1 , e 2 , f 2 .
Theorem 9.5. Let p be an odd prime and let n ≥ 3. Let G be the group generated by {σ, Z, U ′ , V ′ } subject to the relations given below. Then G is isomorphic to Ω + (2n, p). Proof. By (i) we may define the elements Z ij in G.
As in the case e > 1, σ 12 (1), σ 1,−2 (1), Z, U ′ is isomorphic in G to Ω + (4, p). The parenthetical Steinberg relations under each of the headings (a) to (d) all follow from one example, as follows. We may assume, by conjugation, that the relation holds for the given suffices with s = t = 1. We thus deduce that all Steinberg relations hold in G, and complete the proof arguing as in the case e > 1.
9.1.5. A presentation for PΩ + (2n, q). The centre of Ω + (2n, q) is trivial, unless q n ≡ 1 mod 4, when it has order 2. If it is non-trivial, then it is generated by V ′n if n is even, and by (V ′ Z∆) n(q−1)/4 if n is odd. For (V ′ Z∆) n acts on one n-dimensional space containing e 1 by multiplication by ω −2 , and on a complementary n-dimensional space containing f 1 by multiplication by ω 2 . 9.1.6. Standard generators for Ω + (2n, q). We use the following relations to construct standard generators from presentation generators: , and δ ′ = ∆ −1 . The presentation generator Z = ss ′ . 9.2. Even characteristic.
9.2.1. Generators and notation. Let q = 2 e . Let ω be a primitive element of GF(q).
Let n > 1. We take a hyperbolic basis (e 1 , f 1 , . . . , e n , f n ) of the natural module, so that, if Q is the quadratic form that is preserved, then Q(e i ) = Q(f i ) = 0 for all i, and the corresponding bilinear form is defined by e i .f i = 1 and vanishes on all other pairs of basis vectors.
We define the following elements of SO + (2n, q), where 1 ≤ i, j ≤ n and i = j, and s ∈ GF(q).
All these elements of SO + (2n, q) centralise those basis elements that they are not stated to move. All lie in Ω + (2n, q) except for z i . Note that σ 12 (1), δ 1 δ −U 1 , U, V is isomorphic to SL(n, q).
Theorem 9.6. Let n ≥ 3. Let G be the group generated by {Z, U, V } subject to the relations given below. Then G is isomorphic to N 1 .
Proof. Since, for the stated n, U V 2 and V U U V , as permutations of {1, 2, . . . , n}, stand for (3,4) and (3,4, . . . , n), relations (ii) and (iii) allow us to define Z ij in G, and (vii) then implies that Z 12 = Z 21 , so Z ij = Z ji for all i = j. By (v) and (viii) Z 12 commutes with Z 23 and with Z 34 , and hence all Z ij commute with each other, generating an elementary abelian 2-group A by (iv). Finally Z 12 Z 23 = Z 13 by (vi), so Z ij Z jk = Z ik for all distinct i, j, k, and A is generated by {Z 12 , Z 23 , . . . , Z n−1,n } and has rank at most n − 1. But this group has rank n − 1 in N 1 , and the result follows.
Let N = δ, Z, U, V ≤ Ω + (2n, q). It is isomorphic to a subgroup of index 2 in D 2(q−1) ≀ S n . Theorem 9.7. Let q = 2 e with e > 1, and let n ≥ 3. Let G be the group generated by {δ, Z, U, V } subject to the relations given below. Then G is isomorphic to N . Proof. Since U V and V U stand for (2, 3) and (2, 3, . . . , n) respectively, (ii) and (iii) allow us to define δ i in G. Now (iv) implies that δ 1 commutes with δ 2 , and hence the δ i generate an abelian group A, of exponent dividing q − 1 by (v), and of rank at most n. Since A is a group of order (q − 1) n in Ω + (2n, q) this must also be the case in G. In G, the subgroup of N generated by the Z ij normalises A. This is guaranteed by (vi) and (vii), which assert that δ 1 is inverted by Z 12 = Z 21 and centralised by Z 23 . So G/A is isomorphic to a subgroup of index 2 in C 2 ≀ S n . The result follows.
Since n ≥ 3, N acts transitively by conjugation on {σ ij (s) | |i| = |j|, s ∈ GF(q) × }, a set of size 2n(n − 1)(q − 1), as follows. Elements of Z, U, V permute the suffices, acting as signed permutations, where Z ij maps i to −i, and j to −j, and fixes the other suffices.
By (i) we may define the elements Z ij and δ i in G. Since N acts transitively, in Ω + (2n, q), by conjugation on the set of all root elements σ ij (s), where s = 0, we may define σ ij (s) in G by the same conjugation, by (ii).
The parenthetical Steinberg relations under each of the headings (a) to (d) all follow from one example, as in the case of odd q, the suffices all being positive.
All remaining Steinberg relations involving negative suffices may be deduced as in the case of odd q.
The proof that G is generated by the Steinberg generators follows as before.
Recall that Ω + (2n, q) is simple for even q.
As in the case e > 1, σ 12 (1), U, σ 1,−2 (1), Z is isomorphic in G to Ω + (4, 2). The parenthetical Steinberg relations, under each of the headings (a) to (e), all follow from one example, by conjugation. We complete the proof arguing as in the case e > 1.

A presentation for Ω(2n + 1, q)
We may assume that q is odd, since Ω(2n + 1, q) is isomorphic to Sp(2n, q) if q is even.
10.1. Generators and notation. Let q = p e for an odd prime p. Let ω be a primitive element of GF(q).
Let n > 1. We take a basis (e 1 , f 1 , e 2 , f 2 , . . . , e n , f n , w) of the natural module, with symmetric bilinear form defined by e i .f i = f i .e i = 1 for all i, and w.w = −2, and the form vanishes on all other pairs of basis vectors. Note that, with our standard generators, we assume that w.w = −1/2; we have made this change for compatibility with our treatment of Ω − (2n, q) for odd q.
We define the following elements of SO(2n + 1, q), where 1 ≤ i, j ≤ n and i = j, and s ∈ GF(q).
Let N 1 be the subgroup of Ω(2n + 1, q) generated by {z, U ′ , V ′ }, omitting V ′ if n = 2. This is an extension of an elementary abelian 2-group A of rank 2n − 1 by a copy of S n . Here A is the direct product of a subgroup of rank n − 1, generated by U ′2 and its conjugates, with a second subgroup of rank n generated by z and its conjugates.
These relations, together with (v), also imply that z is centralised by all U ′ , V ′conjugates of U ′2 , so we may label the conjugates of z as z i , where 1 ≤ i ≤ n. Relation (vi) states that z = z 1 commutes with z 2 ; hence, by conjugation, all z i commute and, by (iv), generate an elementary abelian 2-group. Thus G is a split extension of a normal subgroup of order at most 2 n by the group U ′ , V ′ of order 2 n−1 n!, by (i). But N 1 , as a subgroup of Ω(2n + 1, q), has order 2 2n−1 n!, and the result follows.
Theorem 10.2. Let q = p e for an odd prime p and e > 1, and let n ≥ 2. Let G be the group generated by {∆, z, U ′ , V ′ }, omitting V ′ if n = 2, subject to the relations given below. Then G is isomorphic to N .
Proof. Suppose first that n ≥ 3. This presentation is very similar to that for N in Theorem 9.2. The subgroups N and N 1 of Ω + (2n, q) have index 2 in the corresponding subgroups of Ω(2n+1, q), the generator Z = Z 12 being replaced by z = z 1 . The presentation for N given in this theorem differs from the presentation in Theorem 9.2 in items (i) and (iv). Item (i) gives a presentation for the enlarged version of N 1 that results from replacing Z by z. Item (iv) asserts that ∆ is centralised by z 3 , and hence, by conjugation, by z 3 , z 4 , . . . , z n , rather than by Z 34 , Z 45 , . . . , Z n−1,n in the case of Ω + (2n, q). This reflects the fact that the centraliser of ∆ in N 1 is twice as large in the case of Ω(2n + 1, q), since both groups N 1 conjugate transitively the same set of elements ∆ ij . The proof now follows that of Theorem 9.2.
The proof for n = 2 is easy.
The centraliser of σ = σ 12 (1) in N is the direct product of the subgroup of N that centralises e 1 , f 1 , e 2 , f 2 with the copy of Q 2(q−1) generated by {∆ 1,−2 , U ′z }. If n = 2 then the centraliser is Q 2(q−1) . If n > 2 then the centraliser has order (q − 1) n−1 2 n−2 (n − 2)!, and index 2(q − 1)n(n − 1) in N ; this corresponds to the fact that N acts transitively by conjugation on the set of elements σ ±i,±j (s), where 1 ≤ i = j ≤ n and s ∈ GF(q) × , and these are equal in pairs. The Steinberg relations for the individual root groups may be deduced from (iv) and (v). We must show that the Steinberg relations that give the commutator of two long root elements, that is, of two elements of the form σ ij (s), are all consequences of (a) to (e). These relations correspond precisely to the Steinberg relations (iv)(a)-(iv)(e) of Theorem 9.4, except for an insignificant change to (iv)(e). Since the torus normaliser N in the case of Ω(2n + 1, q) contains the corresponding torus normaliser in Ω + (2n, q) as a subgroup of index 2, the fact that these five instances of Steinberg relations in Ω(2n + 1, q) suffice to imply all Steinberg relations that give the commutator of two long root elements follows a fortiori from Theorem 9.4. Clearly the Steinberg relations that give the commutator of two short root elements all follow, by conjugation in N , from the one instance (f). Similarly the Steinberg relations that give the commutator of a long and a short root element may be deduced from the instances (g), (h) and (j).
Thus the subgroup H of G generated by the Steinberg generators is isomorphic to Ω(2n + 1, q); that U ′ and V ′ lie in H, which is thus the whole of G, follows by the now familiar argument.
Theorem 10.5. Let p be an odd prime and let n ≥ 2. Let G be the group generated by {σ, τ, z, U ′ , V ′ }, omitting V ′ if n = 2, subject to the relations given below. Then G is isomorphic to Ω(2n + 1, p). Proof. By (i) we may define the elements z i in G.
If n > 2 then N 1 acts transitively by conjugation on {σ ±i,±j (±1) | 1 ≤ i = j ≤ n}; the 8n(n − 1) symbols are equal, as matrices, in pairs. Since N 1 has order 2 2n−1 n!, the centraliser in N 1 of σ has order 2 2n−3 (n − 2)!. Thus this centraliser is generated by the subgroup of N 1 that centralises e 1 , f 1 , e 2 , f 2 , which has order 2 2n−5 (n − 2)! as n > 2, together with U ′2 and U ′z 1 . It follows, from item (ii) of this theorem and the Steinberg relation (vi)(f) of Theorem 10.4, that these elements of N 1 centralise σ in G. Thus the elements σ ij (±1) may be defined in G for all i and j, positive or negative, and hence all elements σ ij (s) for s in GF(p). Now let n = 2. By (ii), we may define the elements τ i (s) in G. Moreover N 1 , as matrices, permutes the 8 elements {σ ±i,±j (±1) | 1 ≤ i = j ≤ 2} in 2 orbits of size 4. Since N 1 has order 16, the centraliser of σ in N 1 has order 4 and is generated by {U ′2 , U ′z 1 }. As in the case n > 2 these elements centralise σ as elements of G, and so we may now define the elements σ ij (s) in this group.
The general cases of the Steinberg relations now follow easily from the cases given in the presentation.
We complete the proof arguing as in the case e > 1. Relation (vi) is needed because the Steinberg presentation gives the universal Chevalley group, and Ω(7, 3) has a Schur multiplier of order 6. The involution in the multiplier is killed by our other relations, and (vi) is needed to kill the remaining 3-cycle. We prove correctness of the presentation for this group by coset enumeration.

11.1.2.
A presentation for Ω − (6, q). We exploit the isomorphism between Ω − (6, q) and SU(4, q)/ −I 4 . Let W be the natural module for SU (4, q). Take a hyperbolic basis (u 1 , v 1 , u 2 , v 2 ) for W and define a basis (e 1 , f 1 , e 2 , f 2 , w 1 , w 2 ) for W ∧ W thus: The natural action of SU(4, q)/ −I 4 on W ∧ W coincides with the natural action of Ω − (6, q), with the basis defined above, under the isomorphism defined below. The kernel of this homomorphism is, of course, −I 4 . Equating an element of SU(4, q) with its image modulo −I 4 , and using our notation for elements of SU(4, q), this homomorphism maps our chosen generators for SU (4, q) to Ω − (6, q) as follows: Theorem 11.1. If q is odd, then a presentation for Ω − (6, q) on our chosen generators is given by taking the presentation for SU(4, q)/ −I 4 in Corollary 7.8 with the above substitutions.
11.1.3. A presentation for Ω − (2n, q) for n ≥ 4. Let N 1 be the subgroup of Ω − (2n, q) generated by {z, U ′ , V ′ }. This is the same generating set as used in Theorem 10.1 for a subgroup of Ω(2n − 1, q), so the same presentation holds.
Let N be the subgroup of Ω − (2n, q) generated by {δ, z, U ′ , V ′ }. Now δ and its conjugates generate a subgroup of N that is the direct product of the cyclic group δ of order (q 2 − 1)/2 with the direct product of n − 2 cyclic groups of order q − 1 generated by δ 1 δ −1 2 , δ 2 δ −1 3 , . . . , δ n−2 δ −1 n−1 . This subgroup contains U ′2 = (δ 1 δ −1 2 ) (q−1)/2 . Thus N has order (q + 1)(q − 1) n−1 2 n−2 (n − 1)!, where a factor of 2 n−1 arises from the z i . Proof. By (i) we may take G to contain N 1 . By (iv) and (vii) it follows that U ′2 commutes with δ, so, using (ii) and (iii), we may assume that δ i ∈ G for all i, and that U ′ , V ′ permutes these elements as unsigned permutations of the suffices. By (vii) it then follows that the δ i all commute. The actions of z 1 and of z 2 on δ 1 are given by (v) and (viii), and this gives the action of z i on δ j for all i and j. If n > 4 it follows from (iii) and (ix), or from (ii) and (ix) if n = 4, that [δ q−1 , V ′ ] = 1. Hence δ q−1 i = δ q−1 j for all i and j. Thus the order of the normal subgroup of G generated by the δ i has order dividing (q − 1) n−1 (q + 1)/2. But N 1 has order 2 2n−3 (n − 1)!, and contains a normal subgroup of order 2 n−2 , namely the subgroup generated by U ′2 and its conjugates, that is contained in the subgroup generated by the δ i . So G has order at most (q + 1)(q − 1) n−1 2 n−2 (n − 1)!, and as this is the order of N the result follows.
Theorem 11.3. The centraliser of σ = σ 12 (1) in N has index 2(n − 1)(n − 2)(q − 1) in N and is generated by the following elements: Proof. The first claim follows from the fact that N permutes transitively the 4(n − 1)(n − 2)(q − 1) elements σ ij (s), for 1 ≤ |i| = |j| ≤ n − 1 and s ∈ GF(q) × , which are equal in pairs. The listed elements all centralise σ, and generate a group G that contains a normal subgroup K of order (q + 1)/2 generated by δ q−1 . Observe that G/K is isomorphic to the direct product of a copy of D 2(q−1) ≀ S n−3 generated by the images of the elements (i) to (iv) with a copy of Q 2(q−1) generated by the images of the elements (v) and (vi). So G has order (q + 1)(q − 1) n−2 2 n−3 (n − 3)!, which is the order of the centraliser.
Theorem 11.10. Let q be even and let n ≥ 4. Let G be the group generated by {σ, τ, δ, z, U, V } subject to the relations given below. Then G is isomorphic to Ω − (2n, q).
(i) Defining relations for N = δ, z, U, V as in Theorem 11.7, but omitting relations (iv) and (v) from Theorem 11.6, relations (iv) to (viii) of Theorem 11.7, and the relation U 2 = 1 from the presentation for U, V . (ii) Relations that state that the elements listed in Theorem 11.8 centralise σ, omitting relations (v) and (vi). (iii) Relations that state that the elements listed in Theorem 11.9 centralise τ , but omitting the relation [τ, z U ] = 1. Proof. The proof of Theorem 11.5 carries over with no significant change.

Verification and access to results
Both the presentations and definitions of their (presentation and standard) generators are publicly available [20] in Magma. We store the relators as straight-line programs on the generating set.
We have run extensive checks to verify their correctness. By evaluating the relations, it is easy to demonstrate that a classical group of specified dimension d and field size q is a quotient of the corresponding finitely-presented group.
For each type, we have a presentation for N , the normaliser of a maximal torus in a split BN pair. For an extensive range of d and q, typically where the corresponding N has order at most 10 9 , we verified using coset enumeration that this presentation defines N . We constructed the permutation action of N on unordered pairs of root elements, confirming that the orbits and representatives are as claimed. Extensive checks were also performed in the classical group: in particular, the claimed (generating sets and structure of) centralisers in N of root elements have been checked. For relevant d ≤ 10 and q ≤ 9 (and sometimes for larger d and smaller q), we also carried out coset enumeration in each finitely-presented group over the preimage of the maximal subgroup of smallest index, and verified that the resulting permutation representation defines (the central quotient of) the claimed classical group.
We observe that our presentations seem particularly well-suited to coset enumeration, often defining precisely the required number of cosets with no redundancy.