On $\sigma$-permutable subgroups of finite groups

Let $\{\sigma_i \mid i\in I\}$ be some partition of the set of all primes $\mathbb{P}$ and let $G$ be a finite group. $G$ is said to be $\sigma$-full if $G$ has a Hall $\sigma_i$-subgroup for all $i$. A subgroup $A$ of $G$ is said to be $\sigma$-permutable in $G$ if $G$ is $\sigma$-full and $A$ permutes with all Hall $\sigma_i$-subgroups $H$ of $G$ (that is, $AH=HA$) for all $i$. In this paper, we give a survey of some recent results on $\sigma$-permutable subgroups of finite groups.