Presentations of finite simple groups: A quantitative approach

There is a constant C 0 C_0 such that all nonabelian finite simple groups of rank n n over F q \mathbb {F}_q , with the possible exception of the Ree groups 2 G 2 ( 3 2 e + 1 ) ^2G_2(3^{2e+1}) , have presentations with at most C 0 C_0 generators and relations and total length at most C 0 ( log ⁡ n +...

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Bibliographic Details
Published in:Journal of the American Mathematical Society 2008-07, Vol.21 (3), p.711-774
Main Authors: Guralnick, R. M., Kantor, W. M., Kassabov, M., Lubotzky, A.
Format: Article
Language:English
Subjects:
Algebra
Commutators
Coordinate systems
Group theory
Integers
Linear transformations
Mathematical congruence
Polynomials
Root systems
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Summary:There is a constant C 0 C_0 such that all nonabelian finite simple groups of rank n n over F q \mathbb {F}_q , with the possible exception of the Ree groups 2 G 2 ( 3 2 e + 1 ) ^2G_2(3^{2e+1}) , have presentations with at most C 0 C_0 generators and relations and total length at most C 0 ( log ⁡ n + log ⁡ q ) C_0(\log n +\log q) . As a corollary, we deduce a conjecture of Holt: there is a constant C C such that dim ⁡ H 2 ( G , M ) ≤ C dim ⁡ M \dim H^2(G,M) \leq C\dim M for every finite simple group G G , every prime p p and every irreducible F p G {\mathbb F}_p G -module M M .
ISSN:0894-0347
1088-6834
DOI:10.1090/S0894-0347-08-00590-0