On $\sigma$-semipermutable subgroups of finite groups

Abstract

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

arXiv comment14 pages