The Characterisitics Of Some Subgroups And The Structure Of Finite Groups

A subgroup H of a finite group G is called a weakly normal subgroup of implies Hx = H. A finite group G is said to be an H* -group if every minimal subgroup of G and each subgroup of G with order 4 is weakly normal in G. In this thesis, the structure of the finite groups all of whose maximal subgroups of even order are H*- groups have been characterized.A subgroup H of a finite group G is said to be weakly S- supplemented in G if there exists a subgroup K of G such that G, where HSG is the maximal S- permutable subgroup of G contained in H. Li Yangming had investigated the structure of finite groups by using weakly S- supplemented subgroups. Some recent results are generalized. In this thesis, we investigate every weakly S- supplemented subgroup influence to subgroup G is solvability, nilpotency, supersolvability.A subgroup H of a finite group G is said to be SS-supplemented in G if there exists a subgroup K of G such that G = HK and H∩K is S- permutable in K. In this thesis, we prove that a finite group G is solvable if every subgroup of prime odd order of G is SS-supplemented in G, and that G is solvable if and only if every sylow subgroup of odd order of G is SS-supplemented in G. These results improve and extend the classical results.We obtain some main results as follows:Theorem 2.1.1 Let G be a finite group of even order. Each maximal subgroups of even order of G is H*- group. Then G is solvable, and one of the following statements is established.(1) G is a supersolvable group.(2) G is a rnininal non H* - group.(3) G = TM, where is sylow 2- subgroup, M is Hall 2’- subgroup, andTheorem 2.2.1 Let G be a finite group. If each subgroup of prime odd order of a group G is weakly S- supplemented. Then G is solvable.Theorem 2.2.7 If each subgroup of prime odd order of a finite group G is weakly S-supplemented, then G’possesses a normal sylow 2- subgroup S such that G’/S is nilpotent.Theorem 2.2.13 Let G be a finite group. Then G is solvable if and only if every sylow of odd order of G is weakly S- supplemented in G.Theorem 2.2.19 Let S be a sylow 2- subgroup of G. If every subgroup of prime odd order of G is complemented in G and every maximal subgroup of S is weakly S- supplemented in G, then G is supersolvable.Theorem 2.3.1 Let G be a finite group. If each subgroup of prime odd order of a group G is SS-supplemented. Then G is solvable.Theorem 2.3.10 If each subgroup of prime odd order of a finite group G is SS-supplemented, then G’ possesses a normal sylow 2- subgroup S such that G’/S is nilpotent.Theorem 2.3.16 Let G be a finite group. Then G is solvable if and only if every sylow of odd order of G is SS- supplemented in G.Theorem 2.3.25 Let S be a sylow 2-subgroup of G. If every subgroup of prime odd order of G is complemented in G and every maximal subgroup of S is SS- supplemented in G, then G is supersolvable.