## 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.)

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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