On weakly-supplemented subgroups and the structure of finite groups

Abstract

A subgroup $$H$$ of a finite group $$G$$ is weakly-supplemented in $$G$$ if there exists a proper subgroup $$K$$ of $$G$$ such that $$G=HK$$ . We prove that a finite group $$G$$ is solvable if it has a maximal subgroup $$M$$ such that every minimal subgroup of $$M$$ is weakly-supplemented in $$G$$ . As an application, some conditions for a finite group to be supersolvable are given.