Normality Of Subgroups And The Structure Of Finite Groups

In the study of finite groups G,some special subgroups or specific sets will be vested certain normality condition,and combined with method of the minimal order counterexample,we obtain some judgments for p-nilpotency and supersolvability of groups G.The thesis is mainly divided into two parts:In part 1,A subgroup K of group G is said to be weakly s-permutable in G,if there exists a subnormal subgroup T of G such that G=KT and K?T?KsG,where K,G is the maximal s-permutable subgroup of G contained in K.By using the weakly s-permutability a subgroup with order pm of a Sylow p-subgroup on the structures of finite groups,some sufficient conditions on p-nilpotency of G are obtained.In part 2,A subgroup H of group G is said to be weakly semipermutable in G,if there exists a subnormal subgroup K of G such that G-HK and H?K?HssG,where HssG is the maximal s-semipermutable subgroup of G contained in H.On the one hand,we using the weakly semipermutability of the maximal subgroup of the Sylow subgroup on the supersolvability of finite groups.some recent results are generalized,some sufficient conditions for a group contained in a saturated formation F are obtained.On the other hand,we by using the weakly s-semipermutability of a set Md(P),a new relation between the set and the structure of finite groups is established successfully,and some sufficient conditions for nilpotency and supersolvability of finite groups are obtained.