Influence of permutizers of subgroups on the structure of finite groups

A group G is said to satisfy the maximal permutizer condition if the permutizer of any maximal subgroup M of G in G, PG(M), is G. In this paper, we characterize the supersolubility of finite groups by using the maximal permutizer condition. We also get some results for when both G/N and N are supersoluble, which implies that G is supersoluble. Our results unify or generalize some known results.