Influence of permutizers of subgroups on the structure of finite groups

A group GG is said to satisfy the maximal permutizer condition if the permutizer of any maximal subgroup MM of GG in GG, PG(M)PG(M), is GG. 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/NG/N and NN are supersoluble, which implies that GG is supersoluble. Our results unify or generalize some known results.