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On the generation gap of a finite group. (English) Zbl 0594.20004
The generation gap of a finite group G, denoted by gap(G), is defined as the difference between d(G), the minimal number of generators of G, and \(d_ G(I(G))\), the minimal number of generators of the augmentation ideal I(G) as a module over the integral group ring \({\mathbb{Z}}G\). The author obtains certain group-theoretical bounds for gap(G). For a subgroup \(S\subset G\) he defines an integer d(G,S) equal to the minimal number of elements of G needed to generate G together with S. Then for any G there is a prime \(p| | G|\) such that for every subgroup H of order prime to p the inequality gap(G)\(\geq d(G)-d(G,H)-1\) holds. If p and G are such that each non-abelian composition factor of G can be generated by two of its Sylow p-subgroups then for any Sylow p-subgroup \(S\subset G\), d(G)-d(G,S)\(\geq gap(G)\). For many groups these inequalities give an estimation of gap(G) up to \(\pm 1\).
Reviewer: S.Jackowski

MSC:
20C05 Group rings of finite groups and their modules (group-theoretic aspects)
20F05 Generators, relations, and presentations of groups
20J06 Cohomology of groups
20D60 Arithmetic and combinatorial problems involving abstract finite groups
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