On $σ$-semipermutable subgroups of finite groups

Abstract

Let $σ=\{σ_{i} | i\in I\}$ be some partition of the set of all primes $\Bbb{P}$, $G$ a finite group and $σ(G) =\{σ_{i} |σ_{i}\cap π(G)\ne \emptyset \}$. A set ${\cal H}$ of subgroups of $G$ is said to be a \emph{complete Hall $σ$-set} of $G$ if every member $\ne 1$ of ${\cal H}$ is a Hall $σ_{i}$-subgroup of $G$ for some $σ_{i}\in σ$ and ${\cal H}$ contains exact one Hall $σ_{i}$-subgroup of $G$ for every $σ_{i}\in σ(G)$. A subgroup $H$ of $G$ is said to be: \emph{$σ$-semipermutable in $G$ with respect to ${\cal H}$} if $HH_{i}^{x}=H_{i}^{x}H$ for all $x\in G$ and all $H_i\in {\cal H}$ such that $(|H|, |H_{i}|)=1$; \emph{$σ$-semipermutable in $G$} if $H$ is $σ$-semipermutable in $G$ with respect to some complete Hall $σ$-set of $G$. We study the structure of $G$ being based on the assumption that some subgroups of $G$ are $σ$-semipermutable in $G$.