Presentations of finite simple groups: A quantitative approach

Guralnick, R.; Kantor, W.; Kassabov, M.; Lubotzky, A.

doi:10.1090/S0894-0347-08-00590-0

Presentations of finite simple groups: A quantitative approach
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by R. M. Guralnick, W. M. Kantor, M. Kassabov and A. Lubotzky

J. Amer. Math. Soc. 21 (2008), 711-774

DOI: https://doi.org/10.1090/S0894-0347-08-00590-0

Published electronically: February 18, 2008

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Abstract:

There is a constant $C_0$ such that all nonabelian finite simple groups of rank $n$ over $\mathbb {F}_q$, with the possible exception of the Ree groups $^2G_2(3^{2e+1})$, have presentations with at most $C_0$ generators and relations and total length at most $C_0(\log n +\log q)$. As a corollary, we deduce a conjecture of Holt: there is a constant $C$ such that $\dim H^2(G,M) \leq C\dim M$ for every finite simple group $G$, every prime $p$ and every irreducible ${\mathbb F}_p G$-module $M$.

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