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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

##### MSC:
 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.)
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##### References:
 [1] Aschbacher, M.; Scott, L., Maximal subgroups of finite groups, J. algebra, 92, 44-80, (1985) · Zbl 0549.20011 [2] Ballester-Bolinches, A., Maximal subgroups and formations, J. pure appl. algebra, 61, 223-232, (1989) · Zbl 0687.20014 [3] Beidleman, J.C.; Spencer, A.E., The normal index of maximal subgroups in finite groups, Illinois J. math., 16, 95-101, (1972) · Zbl 0232.20032 [4] Deskins, W.E., On maximal subgroups, Proc. sympos. pure math., 1, 100-104, (1959) · Zbl 0096.24801 [5] Förster, P., Projektive klassen endlicher gruppen I. schunck-und gaschützklassen, Math. Z., 186, 149-178, (1984) · Zbl 0544.20015 [6] Förster, P., A note on primitive groups with small maximal subgroups, Pub. mat. U.A.B., 28, 19-28, (1984) · Zbl 0584.20007 [7] Förster, P., Projektive klassen endlicher gruppen II, (), Arch. math., 44, 193-209, (1985) · Zbl 0564.20012 [8] Huppert, B., Endliche gruppen I, (1967), Springer Berlin · Zbl 0217.07201 [9] Lafuente, J., Eine note über nichtabelsche hauptfaktoren und maximale untergruppen einer endlichen gruppe, Comm. algebra, 13, 9, 2025-2036, (1985) · Zbl 0575.20020 [10] Mukherjee, N.P., A note on normal index and maximal subgroups in finite groups, Illinois J. math., 75, 173-178, (1975) · Zbl 0303.20014 [11] Mukherjee, N.P.; Bhattacharya, P., The normal index of a finite group, Pacific J. math., 132, 1, 143-149, (1988) · Zbl 0649.20019
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