Presentations of finite simple groups: profinite and cohomological approaches

Guralnick, Robert M.; Kantor, William M.; Kassabov, Martin; Lubotzky, Alexander

doi:10.4171/ggd/22

Cited by 33 publications

(74 citation statements)

References 47 publications

(67 reference statements)

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“…On the other hand, [GKKL2] obtains the constant C = 18, and shows that this is virtually equivalent to the statement that all finite simple groups have profinite presentations with 2 generators and 18 relations. Also [GKKL2,Theorem C] is a generalization of the preceding Corollary to all finite groups (where C becomes 19).…”

Section: Introductionmentioning

confidence: 84%

“…This conjecture has already been proven twice, in [GKKL1,Theorem B ] and [GKKL2,Theorem B]. As in [GKKL1], the conjecture is an immediate consequence of Theorem B (using the elementary result [GKKL1,Lemma 7.1]), except for the Ree groups 2 G 2 (q)-which also had to be handled separately in [GKKL1].…”

Section: Introductionmentioning

confidence: 87%

“…There is also a version for almost quasisimple groups (Corollary 9.4). Both the bounds of 20 profinite relations in [GKKL2] and 51 relations here are not optimal (cf. Remark 4 in Section 11).…”

Section: Introductionmentioning

confidence: 98%

“…In [GKKL2] we proved the existence of profinite presentations for all finite quasisimple groups using fewer than 20 relations. (Recall that a nontrivial group is called quasisimple if it is both perfect and simple modulo its center.…”

Section: Introductionmentioning

confidence: 99%

“…Also [GKKL2,Theorem C] is a generalization of the preceding Corollary to all finite groups (where C becomes 19).…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Introductionmentioning

confidence: 84%

Section: Introductionmentioning

confidence: 87%

Section: Introductionmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

“…Also [GKKL2,Theorem C] is a generalization of the preceding Corollary to all finite groups (where C becomes 19).…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Presentations of finite simple groups: a computational approach

Guralnick¹,

Kantor²,

Kassabov³

et al. 2010

J. Eur. Math. Soc.

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Bounding the Dimensions of Rational Cohomology Groups

Bendel

Boe

Drupieski

et al. 2014

Developments and Retrospectives in Lie Theory

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Let k be an algebraically closed field of characteristic p > 0, and let G be a simple simply-connected algebraic group over k that is defined and split over the prime field Fp. In this paper we investigate situations where the dimension of a rational cohomology group for G can be bounded by a constant times the dimension of the coefficient module. We then demonstrate how our results can be applied to obtain effective bounds on the first cohomology of the symmetric group. We also show how, for finite Chevalley groups, our methods permit significant improvements over previous estimates for the dimensions of second cohomology groups.

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Polynomial-Time Isomorphism Test of Groups that are Tame Extensions

Grochow

Qiao

2015

Algorithms and Computation

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We give new polynomial-time algorithms for testing isomorphism of a class of groups given by multiplication tables (GpI). the celebrated tame-wild dichotomy in representation theory. We then solve new cases of GpI in polynomial time.Our result relies crucially on the divide-and-conquer strategy proposed earlier by the authors (CCC 2014), which splits GpI into two problems, one on group actions (representations), and one on group cohomology. Based on this strategy, we combine permutation group and representation algorithms with new mathematical results, including bounds on the number of indecomposable representations of groups in the tame case, and on the size of their cohomology groups.Finally, we note that when a group extension is not tame, the preceding bounds do not hold. This suggests a precise sense in which the tame-wild dichotomy from representation theory may also be a dividing line between the (currently) easy and hard instances of GpI. *

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Presentations of finite simple groups: profinite and cohomological approaches

Abstract: Abstract. We prove the following three closely related results:(1) Every finite simple group G has a profinite presentation with 2 generators and at most 18 relations. (2) If G is a finite simple group, F a field and M an F G-module, then dim

Cited by 33 publications

References 47 publications

Presentations of finite simple groups: a computational approach

Presentations of finite simple groups: a computational approach

Bounding the Dimensions of Rational Cohomology Groups

Polynomial-Time Isomorphism Test of Groups that are Tame Extensions

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