## 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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