Weakly S-Semipermutable Subgoups And SS-Supplement Subgroups

The composition of group has a great relationship with the character of some subgroup, investigating the characters of different subgroup. We can get group which have different compositions. A subgroup H of a finite group G is called s-semipermutable if it is permutable with every Sylow p-subgroup of G with (p,|H|)= 1. A sub group H of a finite group G is called weakly s-semipermutable in G if there is a subgroup T of G such that G=HT and H∩T≤HSSG, where HSSG is the subgroup of H generated by all those subgroup of H which are s-semipermutable in G. A subgroup H of a finite group G is called ss-supplement in G if there is a subgroup T of G such that HT is a subgroup of G which are s-permutable in G and H∩T≤HSG, where HSG is the subgroup of H generated by all those subgroup of H which are s-permutable in G. In this paper, by using weakly s-semipermutability and ss-supplement property of some special subgroups (such as cyclic subgroups of Sylow subgroups, maximal subgroups of Sylow subgroups) of G, we obtain some sufficient condition for a finite group to be p-nilpotent, supersolvable. Some of the previous results are generalized.The thesis is divided into three parts according to its contents.In part 1, we introduce the investigative background of this paper and give a brief account of our main results.In part 2, set out some basic concepts and lemmas which are closely related to our results.In part 3, we use the character of weakly s-semipermutable subgroups and ss-supplement subgroups, obtain some sufficient conditions for p-nilpotent and supersolvablity of a finite group.