Presentations of finite simple groups: A quantitative approach

Guralnick, R.¹ ; Kantor, W.² ; Kassabov, M.³ ; Lubotzky, A.⁴

¹ Department of Mathematics, University of Southern California, Los Angeles, California 90089-2532
² Department of Mathematics, University of Oregon, Eugene, Oregon 97403
³ Department of Mathematics, Cornell University, Ithaca, New York 14853-4201
⁴ Department of Mathematics, Hebrew University, Givat Ram, Jerusalem 91904, Israel

Journal of the American Mathematical Society, Tome 21 (2008) no. 3, pp. 711-774

Résumé

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 $M$.

Détail

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

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