## Maximal subgroups of finite groups.(English)Zbl 0549.20011

The main result of this paper is theorem 1, which gives a general structure of a finite group $$G$$ with non-empty set $$n^*_G$$ of maximal subgroups $$M$$ such that $$\cap\{M^ g\mid g\in G\}=1,$$ and reduces the problem of determination of the set of conjugacy classes of elements of $$n^*_G$$ to the similar problem for groups of the shape $$VH$$, the semidirect product of a faithful irreducible $$H$$-module $$V$$ over a field of prime order and a finite group $$H$$ with $$L\leq H\leq \operatorname{Aut}(L)$$ for some nonabelian simple group $$L$$.
In the process of obtaining theorem 1 two results (theorems 2 and 3) concerning 1-cohomology are obtained. For example, theorem 3 determines the structure of the generalized Fitting subgroups of a finite group $$G$$ if $$H^1(G,V)\neq 0$$ for a faithful irreducible $$G$$-module $$V$$ over some field of prime characteristic, and describes the representation of that group on $$V$$.
Finally, some corrections to the paper of L. Scott [Proc. Symp. Pure Math. 37, 319–331 (1980; Zbl 0458.20039)] are given.

### MSC:

 20D05 Finite simple groups and their classification 20D25 Special subgroups (Frattini, Fitting, etc.) 20D30 Series and lattices of subgroups 20J05 Homological methods in group theory

Zbl 0458.20039
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### References:

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