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Zbl 0804.20015
Carocca, Angel
$p$-supersolvability of factorized finite groups. (English)
[J] Hokkaido Math. J. 21, No.3, 395-403 (1992). ISSN 0385-4035

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 $G = HK \ne 1$ and $H$ and $K$ are mutually permutable, then $H$ or $K$ contains a nontrivial normal subgroup of $G$ or $F(G) \ne 1$ (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.
[H.Heineken (Würzburg)]

MSC 2000:: *20D40 Products of subgroups of finite groups
20D10 Solvable finite groups
20D20 Sylow subgroups of finite groups

Keywords: mutually permutable subgroups; normal subgroup; $p$-supersoluble; $p$- nilpotent

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