The influence of supplemented subgroups on the structure of finite groups

Abstract

A subgroup H of a finite group G is said to be supplemented in G if there exists a subnormal subgroup T of G such that G = HT and where is the Frattini subgroup of H. In this article, we investigate the structure of a finite group G under the assumption that certain subgroups of G are supplemented in G. We obtain that a finite group G is nilpotent if and only if every Sylow subgroup of G is supplemented in G. And a group G is nilpotent if every maximal subgroup of G is supplemented in G. Moreover, the commutativity of G is characterized by using that the minimal subgroups of two non–conjugate maximal subgroups of G are supplemented in G.