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 \(G\).