Properties Of Subgroups And The Structures Of Finite Groups

It is a classic and important subject in theory of finite groups to characterize group structures from some properties of subgroups, which is also an important means to study finite groups.Subgroups have many classic properties such as commutation, normality and complement etc. In this thesis, we consider the interplay of the properties of subgroups and the structures of finite groups from two sides.Part one Finite groups in which every subgroup is abelian or normalThe structures of inner-abelian group which is a non-abelian group of every subgroup being abelian have been studied detailed in document [1].Also Hamilton group which is a non-abelian group of every subgroup being normal has already been analyzed in document [2].In this chapter, we investigate the structures of the groups which every subgroup is abelian or normal. Here we give a conception.We write G = [F]H if group G is a semi-direct product with its normal subgroup F and subgroup H.A Probenius group G = [F]H with kernal F and cyclic complemented subgroup H is called a (*)-Frobenius group if F is always an irreducible H₁-subgroup for all 1< H₁≤H.We make two conclusions as follows:Conclusion 1. Let G be a non-nilpotent group. Then every subgroup of G is abelian or normal in G if and only if G/Z(G) is a (*)-Frobenius group and G satisfies one of the following properties:(1) F(G) is abelian where F(G) is the Fitting subgroup of G;(2) F(G) is non-abelian,and F(G) = P×Q where P is a non-abelian Sylow p-subgroup of F(G) and an extra-special p-group with order p³ . In paticular, G/Q = SL(2,3) if p = 2.Conclusion 2. Let G be a non-abelian nilpotent group. If every subgroup of G is abelian or normal in G,then G = P×A with P a non-abelian Sylow p-subgroup and A an abelian p'-Hall subgroup. And more , there exists an abelian normal subgroup TV in P such that every subgroup of P/N is normal in P/N. Part two Influence on super-solublity of finite group of weakly quasi-mormal of some special subgroupsIt brings great influence on the structures of finite groups for the normality,sub-normality, c-normality and quasi-normality etc. In the chapter, we get some sufficient conditions for super-solublity of finite groups by using of the conception " weakly quasi-normality ".A subgroup H of group G is called a weakly quasi-normal subgroup or weakly quasi-normal in G if there exists at least a conjugate subgroup K^x with x∈G for all K≤G such that HK^x = K^xH.We make the following conclusions:Conclusion 3. A finite group G is super-soluble if G satisfies one of the following conditions:(1) A maximal and cyclic subgroup of G is weakly quasi-normal in G;(2) Let M be a subgroup of G with index power of a prime. And all Sylow subgroups and all maximal subgroups of Sylow subgroups of M are weakly quasi-normal in G;(3) Let G be soluble and M be a maximal subgroup of G.And all maximal subgroups of M are weakly quasi-normal in G;(4) All cyclic subgroups of Sylow subgroups of G which is soluble are weakly quasi-normal in G;(5) All maximal subgroups of Sylow subgroups of G which is soluble are weakly quasi-normal in G;(6) Let G be the product of A which is aÏ€-Hall subgroup and B aÏ€'-Hall sub-group.And all Sylow subgroups of A and B are weakly quasi-normal in G.