Generation of almost simple groups
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.
scientific article; zbMATH DE number 826746