On $\sigma$-semipermutable subgroups of finite groups
Let $\sigma =\{\sigma_{i} | i\in I\}$ be some partition of the set of all primes $\Bbb{P}$, $G$ a finite group and $\sigma (G) =\{\sigma_{i} |\sigma_{i}\cap \pi (G)\ne \emptyset \}$. A set ${\cal H}$ of subgroups of $G$ is said to be a \emph{complete Hall $\sigma $-set} of $G$ if every member $\ne 1$ of ${\cal H}$ is a Hall $\sigma_{i}$-subgroup of $G$ for some $\sigma_{i}\in \sigma $ and ${\cal H}$ 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 ${\cal H}$} 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{${\sigma}$-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 $G$.