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Generation of simple groups. (English) Zbl 0601.20013
The main results of this paper are: Theorem A. Any finite nonabelian simple group can be generated by an involution and a Sylow 2-subgroup. - Theorem B. Let G be a finite group and K a field of characteristic p. If G acts faithfully on the irreducible KG-module V, then dim $$H^ 1(G,V)\leq (2/3)\dim V$$. - Theorem C. Let G be a finite group of even order. Let O(G) be the maximal normal subgroup of G of odd order, and set $$\bar G=G/O(G)$$. Then either G has a maximal subgroup of even index or $$A=O_ 2(\bar G)=\Phi (\bar G)$$ and $$G/A\cong A_ 7$$. - Theorem D. Let p be a prime and G a finite group. Then $$G=<P,R>$$ for some p-subgroup P and p’-subgroup R.
The proofs of these results invoke the classification of finite simple groups. Various consequences of these results improve and extend work of several authors.
Reviewer: M.E.Harris

##### MSC:
 20D06 Simple groups: alternating groups and groups of Lie type 20F05 Generators, relations, and presentations of groups
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