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.


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
Full Text: DOI


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