Geodesic
Criteria of $p$-supersolubility of finite groups
Problemy fiziki, matematiki i tehniki, no. 2 (2015), pp. 56-61.

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Let $G$ be a finite group and $H$ a subgroup of $G$. We say that $H$ is $\tau$-quasinormal in $G$ if $H$ permutes with all Sylow subgroups $Q$ of $G$ such that $(|Q|, |H|)=1$ and $(|H|, |Q^G|)\ne1$. The main result here is the following: Let $G=AT$, where $A$ is a Hall $\pi$-subgroup of $G$ and $T$ is $p$-nilpotent for some prime $p\notin\pi$, let $P$ denote a Sylow $p$-subgroup of $T$ and assume that $A$ is $\tau$-quasinormal in $G$. Suppose that there is a number $p^k$ such that $1$ and $A$ permutes with every subgroup of $P$ of order $p^k$ and with every cyclic subgroup of $P$ of order $4$ (if $p^k=2$ and $P$ is non-abelian). Then $G$ is $p$-supersoluble.
Keywords: $\tau$-quasinormal subgroup, Sylow subgroup, Hall subgroup, $p$-supersoluble group.
Mots-clés : $p$-soluble group 
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V. O. Lukyanenko; T. V. Tikhonenko. Criteria of $p$-supersolubility of finite groups. Problemy fiziki, matematiki i tehniki, no. 2 (2015), pp. 56-61. http://geodesic.mathdoc.fr/item/PFMT_2015_2_a8/

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