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 PFMT, 2015, Issue 2(23), Pages 56–61 (Mi pfmt374)

MATHEMATICS

Criteria of $p$-supersolubility of finite groups

V. O. Lukyanenko, T. V. Tikhonenko

P. Sukhoi Gomel State Technical University, Gomel, Belarus

Abstract: 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 $1<p^k<|P|$ 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, $p$-soluble group, $p$-supersoluble group.

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UDC: 512.542

Citation: V. O. Lukyanenko, T. V. Tikhonenko, “Criteria of $p$-supersolubility of finite groups”, PFMT, 2015, no. 2(23), 56–61

Citation in format AMSBIB
\Bibitem{LukTik15}
\by V.~O.~Lukyanenko, T.~V.~Tikhonenko
\paper Criteria of $p$-supersolubility of finite groups
\jour PFMT
\yr 2015
\issue 2(23)
\pages 56--61
\mathnet{http://mi.mathnet.ru/pfmt374}