Conditionally Permutable Subgroups and Supersolubility of Finite Groups

Abstract

A subgroups H of a group G is called conditionally permutable (or in brevity, c-permutable) subgroup in G if for every subgroup T of G there exists an element x ∈ G such that HT ͯ = Tˣ H. If H is c-permutable in every subgroup of G containing H, then H is said to be completely c-permutable in G. Our main result in this paper is to give a new characterization theorem for supersoluble finite groups by using their c-permutable subgroups. We will prove that a finite group G is supersoluble if and only if every maximal subgroup of G is c-permutable in G. Also, we will prove that a finite group G is supersoluble if every 2-maximal subgroup of G is completely c-permutable in G. In addition, we will determine the structure of a finite soluble group G in which every subnormal subgroup is c-permutable with all Sylow subgroups.