On $\sigma$-quasinormal subgroups of finite groups

Abstract

Let $G$ be a finite group and $\sigma =\{{\sigma }_i|i\in I\}$ some partition of the set of all primes $\mathbb{P}$, that is, $\sigma =\{{\sigma }_i|i\in I\}$, where $\mathbb{P}={\bigcup }_{i\in I}{\sigma }_i$ and ${\sigma }_i\cap {\sigma }_j=\varnothing$ for all $i\ne j$. We say that $G$ is $\sigma$-primary if $G$ is a ${\sigma }_i$-group for some $i$. A subgroup $A$ of $G$ is said to be: $\sigma$-subnormal in $G$ if there is a subgroup chain $A=A_0\le A_1\le \cdots \le A_n=G$ such that either $A_{i-1}⊴A_i$ or $A_i/(A_{i-1}{)}_{A_i}$ is $\sigma$-primary for all $i=1,\dots ,n$, modular in $G$ if the following conditions hold: (i) $⟨X,A\cap Z⟩=⟨X,A⟩\cap Z$ for all $X\le G,Z\le G$ such that $X\le Z$, and (ii) $⟨A,Y\cap Z⟩=⟨A,Y⟩\cap Z$ for all $Y\le G,Z\le G$ such that $A\le Z$. In this paper, a subgroup $A$ of $G$ is called $\sigma$-quasinormal in $G$ if $L$ is modular and $\sigma$-subnormal in $G$. We study $\sigma$-quasinormal subgroups of $G$. In particular, we prove that if a subgroup $H$ of $G$ is $\sigma$-quasinormal in $G$, then for every chief factor $H/K$ of $G$ between $H^G$ and $H_G$ the semidirect product $(H/K)⋊(G/C_G(H/K))$ is $\sigma$-primary.

9 pages