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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
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