## $$p$$-supersolvability of factorized finite groups.(English)Zbl 0804.20015

The author calls two subgroups $$H$$, $$K$$ of a group mutually permutable if $$H$$ is permutable with every subgroup of $$K$$ and $$K$$ is permutable with every subgroup of $$H$$. He obtains the following main results: If $$G = HK \neq 1$$ and $$H$$ and $$K$$ are mutually permutable, then $$H$$ or $$K$$ contains a nontrivial normal subgroup of $$G$$ or $$F(G) \neq 1$$ (Theorem A). If $$G = HK$$ and $$H$$ and $$K$$ are $$p$$-supersoluble and mutually permutable, if further $$G'$$ is $$p$$-nilpotent, then $$G$$ is $$p$$-supersoluble (Theorem B). – If $$G = HK$$, where $$H$$ and $$K$$ are mutually permutable, $$H$$ is $$p$$- supersoluble and $$K$$ is $$p$$-nilpotent, then $$G$$ is $$p$$-supersoluble.

### MSC:

 20D40 Products of subgroups of abstract finite groups 20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, $$\pi$$-length, ranks 20D20 Sylow subgroups, Sylow properties, $$\pi$$-groups, $$\pi$$-structure
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