On weakly $s$-permutably embedded subgroups of finite

Authors

  • Yanming Wang Sun Yatsen University

Keywords:

weakly s-permutably embedded, the generalized Fitting subgroup, $p$-nilpotent group, saturated formation.

Abstract

A subgroup $H$ is called weakly s-permutably embedded in $G$ if there are a subnormal subgroup $T$ of $G$ and an s-permutably embedded subgroup $H_{se}$ of $G$ contained in $H$ such that $G=HT$ and $H\cap T \leq H_{se}$. In this note, we study the influence of weakly $s$-permutably embedded property of subgroups on the structure of $G$, and obtain the following theorem: Let $\cal F$ be a saturated formation containing ${\cal U}$, the class of all supersolvable groups, and $G$ a group with $E$ as a normal subgroup of $G$ such that $G/E\in\cal F$. Suppose $P$ has a subgroup $D$ such that $1<|D|<|P|$ and all subgroups $H$ of $P$ with order $|H|=|D|$ are $s$-permutably embedded in $G$. When $p= 2$ and $|D|=2$, in addition, we suppose that each cyclic subgroup of $P$ of order $4$ is weakly $s$-permutably embedded in $G$. Then $G\in\cal F$. DOI: 10.1017/S0004972712000238

Author Biography

Yanming Wang, Sun Yatsen University

Lingnan College

Published

2012-06-25

Issue

Vol. 86 No. 1 (2012)

Section

Articles
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