The influence of complemented minimal subgroups on the structure of finite groups

Abstract

A subgroup H of a ¯nite group G is said to be complemented in G if there exists a subgroup K of G such that G = HK and H K = 1. In this paper the following theorem is proved: Let G be a ¯nite group and let p be the smallest prime dividing the order of G. Then G is p-nilpotent if and only if every minimal subgroup of P GN is complemented in NG(P), where P is a Sylow p-subgroup of G and GN is the nilpotent residual of G. As some applications, some interesting results related with complemented minimal subgroups are obtained.