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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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