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On the normal index of maximal subgroups in finite groups. (English) Zbl 0699.20016
The author uses results from primitive group theory to generalize some work of Beidleman and Spencer, Mukherjee, and Mukherjee and Bhattacharya on criteria for solvability, supersolvability, and nilpotency to \(\Pi\)- solvability, \(\Pi\)-supersolvability, and \(\Pi\)-nilpotency. Typical is Theorem 1: Let \(\Pi\) be a set of primes. If finite group G has a \(\Pi\)- solvable maximal subgroup M whose normal index equals its index in G, then G is \(\Pi\)-solvable.
Reviewer: W.E.Deskins

20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks
20E28 Maximal subgroups
20D20 Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure
20D25 Special subgroups (Frattini, Fitting, etc.)
Full Text: DOI
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