The Influence Of Certain Properties Of Subgroups On The Structure Of Finite Groups

When we do research about the structure of a group,the influence of subgroups is important.We can get a lot of information about the structure of the group from the properties of its subgroups.In this paper,the main work is to study the structure of a finite group on the basis of [1],[2],[3]. This paper is composed of three chapters.In chapter 1 ,on one hand ,we give some sufficient conditions for a finite group to be supersolvable and nilipotent by using the properties of π-supplemented subgroups;For example: Theorem 3 Let G be a group, 2 ∈ π, if every subgroup of G of prime order is contained in SE(G), every cyclic subgroup of G of order 4 is π-supplemented in G, then G will be supersolvable.Theorem 7 Let G be a group, if every subgroup of G of prime order is contained in Z_∞(G),2 ∈π, every cyclic subgroup of G of order 4 is π-supplemented in G,then G will be nilpotent.On the other hand we study the influence of π-supplemented subgroups on formation.For example: Theorem 10 Let T be a subgroup-closed local formation with the following properties :an inner F-group which is solvable and its F-residual is a Sylow subgroup .2∈6 π, if every cyclic subgroup of G of order 4 is π-supplemental in G and every minimal subgroup of G is contained in the F-hypercenter of G ,then G is an F-group.In chapter 2, on one hand ,we give some sufficient conditions for a finite group to be supersolvable by using the properties of conditional permutable and completely conditional permutable between subgroups;For example: Theorem 4 If every normalizer of Sylow subgroup of G is completely conditional permutable,then G will be a supersolvable group.On the other hand ,we give some sufficient conditions for products of two subgroups to be supersolvable by using conditional pemutable between subgroups.For example: Theorem 12 Let H, Kbe supersolvable subgroup of G, G = HK, G'is a nilpotent group ,if H is conditional permutable in K, if K is conditional permutable in i/,then G will be a supersolvable group.In chapter 3, we generalize two theorems in [3],obtain some more profound results.For example: Theorem 1 LetG be a finite group, p is the smallest prime divisor of \G\,P G Sylp(G),suppose P is an Abel group,other prime divisor of \G\ is larger than pn,then G will be a p-nilpotent group.