X-permutable Subgroups and Supersolubility of Finite Groups

Using the concept,X-permutable subgroup of finite groups,two sufficient conditions of supersoluble groups are obtained:(1) G is a soluble group,let X be a subgroup of G such that X contains every maximal subgroup and every minimal subgroup of G.Suppose that every maximal subgroup of G is X-permutable with every maximal subgroup of every Sylow subgroup of G in G,then G is a supersoluble group;(2)Let K be a normal subgroup of G,let X be a subgroup of G such that X contains every p-subgroup of G.Suppose that every maximal subgroup of G not containing K is X-permutable in G,then K is a supersoluble group.