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On theta pairs for maximal subgroups. (English) Zbl 0830.20045

The author pursues the study of \(\theta\)-pairs of maximal subgroups of finite groups, a variation on normal index introduced by N. P. Mukherjee and P. Bhattacharya [Proc. Am. Math. Soc. 109, 589-596 (1990; Zbl 0699.20019)], and establishes necessary and sufficient conditions for a group to be \(\pi\)-solvable, \(\pi\)-supersolvable and \(\pi\)-nilpotent. For example, for a set of primes, \(\pi\), a finite group \(G\) is \(\pi\)-solvable if and only if each \(c\)-maximal subgroup of \(G\) (a maximal subgroup of composite index) has a normal maximal \(\theta\)-pair \((A,B)\) with \(A/B\) either \(\pi\)-solvable or a \(\pi'\)-group.

MSC:

20D30 Series and lattices of subgroups
20D20 Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure
20E28 Maximal subgroups
20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks
20D25 Special subgroups (Frattini, Fitting, etc.)

Citations:

Zbl 0699.20019
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References:

[1] DOI: 10.1016/0022-4049(90)90150-G · Zbl 0699.20016
[2] DOI: 10.1016/0022-4049(86)90074-5 · Zbl 0597.20014
[3] Gorenstein D., Finite Groups (1968)
[4] Huppert B., Endliche gruppen 1 (1967)
[5] Mukherjee, N. P. and Bhattacharya, P. On theta pairs for a maximal subgroup. Proc. Amer. Math. Soc. Vol. 109, pp.585–596. · Zbl 0699.20019
[6] Mukherjee N. P., Pacific J. Math 132 pp 143– (1988)
[7] Weinstein M., Between nilpotent and solvable (1982)
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