X-可换子群与有限群的超可解性

利用X-可换子群的概念,得到了有限群超可解的2个充分条件:(1)设G是可解群,X是G的子集且包含G的极小子群和极大子群。如果G的每个极大子群和G的sylow子群的每个极大子群在G中X-可换,那么G是超可解群;(2)设K■G,X是G的子集且包含G的p-子群。如果每个不包含K的G的极大子群在G中X-可换,那么K是超可解群。

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.