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Some questions on the number of generators of a finite group. (English) Zbl 0734.20012
Let d(G) and \(d_ p(G)\) denote the minimum number of generators of a finite group G and of its p-Sylow subgroup respectively. Then in a previous publication the author showed that \[ d(G)\leq \max_{p} d_ p(G)+1=d+1. \] He now proceeds with a further investigation of this inequality. He derives consequences from the assumptions that the above inequality is an equality and \(d(G)=d_{ZG}(I_ G)\), where the latter number is the minimal number of generators of the augmentation ideal \(I_ G\) as a ZG-module. If on the other hand \(d(G)\neq d_{ZG}(I_ G)\), then \(d(G)\leq d_ 2(G)+1\). Finally it is shown that if G is perfect, then d(G)\(\leq d\) and \(d(G)\leq \max \{\frac12d+2,d_ 2(G)\}\).

MSC:
20F05 Generators, relations, and presentations of groups
20D60 Arithmetic and combinatorial problems involving abstract finite groups
20D20 Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure
20C05 Group rings of finite groups and their modules (group-theoretic aspects)
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