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On maximal subgroups of finite groups and theta pairs. (English) Zbl 0892.20021

This paper presents further investigations on \(\theta\)-pairs and completions in finite groups, concepts introduced by N. Mukherjee and P. Bhattacharya and by the reviewer. In the first section the authors exhibit the relationship between completions and \(\theta\)-pairs. Then they obtain necessary and sufficient conditions for solvability in terms of \(\theta\)-pairs (Theorems 2 and 4) and they study the intrinsic properties of a given maximal subgroup \(M\) of finite group \(G\) and its associated \(\theta\)-pairs which imply \(G\) to be solvable, \(\pi\)-solvable or supersolvable (Theorems 1, 3 and 5).

MSC:

20E28 Maximal subgroups
20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks
20D25 Special subgroups (Frattini, Fitting, etc.)
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