A characterization of solubility of finite groups

Abstract

A subgroup H of a group G is said to be G-semipermutable in G if H has a supplement T in G such that $$G=HT$$ and for every subgroup $$T_1$$ of T, $$HT_1^g=T_1^gH$$ for some element g in G. In this paper, we investigate the structure of G under the assumption that a soluble maximal subgroup of G is G-semipermutable in G.