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Generation of almost simple groups. (English) Zbl 0839.20021

Let \(d(X)\) denote the minimal number of generators for a group \(X\). It is well known that any finite non-abelian simple group is 2-generated. In this paper, the authors prove that if \(G\) is a finite almost simple group, i.e. \(S\leq G\leq \text{Aut} (S)\) for a finite simple non-abelian group \(S\), then \(d(G)= \max\{2, d(G/S)\}\leq 3\) and \(d(G) =3\) if and only if the group \(Z_2\times Z_2\times Z_2\) is an epimorphic image of \(G\) and, in particular, \(S\) is isomorphic to \(A_{2n+1} (p^h)\) or \(D_n (p^h)\), where \(p\) is odd and \(h\) is even. By the same token they give an answer to Gruenberg’s problem 11.27 from “The Kourovka notebook” [Novosibirsk (1992; Zbl 0831.20003)]. Finally, some consequences about the presentation rank of a finite group are given.

MSC:

20D05 Finite simple groups and their classification
20F05 Generators, relations, and presentations of groups

Citations:

Zbl 0831.20003
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