Finite group with given $c$-permutable subgroups

Abstract

Following [1] we say that subgroups  $H$ and $T$ of a group $G$ are $c$- permutable in $G$ if there exists an element $x\in G$ such that $HT^x=T^xH$.  We prove that a finite soluble group $G$ is supersoluble if and only if every maximal subgroup of every Sylow subgroup of $G$ is c-permutable with all Hall subgroups of $G$.