The efficiency of simple groups of order &lt; 10<sup>5</sup>

Campbell, C. M.; Robertson, E. F.

doi:10.1080/00927878208822711

Cited by 20 publications

(39 citation statements)

References 8 publications

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“…Suitable presentations for the groups SL 2 (p) are given in [9] and for L 3 (3) and the full covering group of Sz(8) in [7,8] 2. Thus, we can apply Proposition 2, with r = 2.…”

Section: Commutators In Minimal Non-soluble Groups Partmentioning

confidence: 99%

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Finite Axiomatization of Finite Soluble Groups

Wilson¹

2006

J. London Math. Soc.

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“…Suitable presentations for the groups SL 2 (p) are given in [9] and for L 3 (3) and the full covering group of Sz(8) in [7,8] 2. Thus, we can apply Proposition 2, with r = 2.…”

Section: Commutators In Minimal Non-soluble Groups Partmentioning

confidence: 99%

“…Since [ker ε, E] = 1 and [K, G] = K we have [S 1 , F ] R and [S 1 , F ]S = S 1 , and we conclude that S 1 = RS. Thus, d G (K) = d F (RS/S) d F (R) r, as required.We shall make use of the results of Campbell and Robertson[7][8][9] on presentations of full covering groups of certain simple groups.…”

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confidence: 99%

Finite Axiomatization of Finite Soluble Groups

Wilson¹

2006

J. London Math. Soc.

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“…An efficient presentation of PSL (2,3 2 ) = A 6 is given in [6]. Efficient presentations of PSL (2,3 3 ), PSL (2,5 2 ) and PSL (2,I 2 ) are given in [8], see also [7].…”

Section: Suppose P" = -1 (Mod 4) a Is A Primitive Element Of Gf(p") mentioning

confidence: 99%

On presentation of PSL (2, pⁿ)

1990

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We give presentations for the groups PSL(2,p"), p prime, which show that the deficiency of these groups is bounded below. In particular, for p = 2 where SL(2,2") = PSL(2,2"), we show that these groups have deficiency greater than or equal to -2 . We give deficiency -1 presentations for direct products of SL(2,2"') for coprime n,-. Certain new efficient presentations are given for certain cases of the groups considered.

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“…If X | R is a presentation of a finite group G, then |R| − |X| is at least the smallest number d(M) of generators of the Schur multiplier M of G; and X | R is called an efficient presentation if |R| − |X| = d(M) [CRKMW,CHRR1,CHRR2]. For 35 years the groups PSL(2, p) with p ≥ 5 prime have contained the only infinite family of finite nonabelian simple groups known to have efficient presentations ( [Sun,CR3]; cf. (3.19)).…”

Section: Introductionmentioning

confidence: 99%

Presentations of finite simple groups: a computational approach

Guralnick¹,

Kantor²,

Kassabov³

et al. 2010

J. Eur. Math. Soc.

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Abstract. All finite simple groups of Lie type of rank n over a field of size q, with the possible exception of the Ree groups 2 G 2 (q), have presentations with at most 49 relations and bit-length O(log n + log q). Moreover, A n and S n have presentations with 3 generators, 7 relations and bitlength O(log n), while SL(n, q) has a presentation with 6 generators, 25 relations and bit-length O(log n + log q).

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The efficiency of simple groups of order < 10⁵

Cited by 20 publications

References 8 publications

Finite Axiomatization of Finite Soluble Groups

Finite Axiomatization of Finite Soluble Groups

On presentation of PSL (2, pⁿ)

Presentations of finite simple groups: a computational approach

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The efficiency of simple groups of order < 105

Cited by 20 publications

References 8 publications

Finite Axiomatization of Finite Soluble Groups

Finite Axiomatization of Finite Soluble Groups

On presentation of PSL (2, pn)

Presentations of finite simple groups: a computational approach

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The efficiency of simple groups of order < 10⁵

On presentation of PSL (2, pⁿ)