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On theta pairs for a maximal subgroup. (English) Zbl 0814.20016

The author continues the investigation of finite groups whose maximal subgroups have nice “\(\theta\)-pairs”, terminology introduced by Mukherjee and Bhattacharya and a variation on the concept of “index complex” introduced by the reviewer. It is shown that \(G\) is solvable if and only if each maxial subgroup \(M\) of \(G\) has a \(\theta\)-pair \((C,D)\) such that \(C/D\) is nilpotent. Furthermore, if \(G\) is \(S_ 4\)-free, then \(G\) is supersolvable if and only if each \(M\) has a \(\theta\)-pair \((C,D)\) with \(C/D\) cyclic.

MSC:

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