X-semipermutable subgroups of finite groups

Let X be a non-empty subset of a group G. Then we call a subgroup A of G a X-semipermutable subgroup of G if A has a supplement T in G such that for every subgroup T1 of T there exists an element x X such that . In this paper, we study the properties of X-semipermutable subgroups. In particular, a new version of the famous Schur–Zassenhaus Theorem in terms of X-semipermutable subgroups is given.