The Influence Of The Properties Of Even Order Maximal Subgroups On The Structure Of Finite Groups

There is an important influence of the properties of subgroups on the structure of finite groups.Studying the structure of finite groups through some properties of subgroups is not only a hot spot,but also a difficult point in group theory.In the light of the Feit-Thompson theorem on solvability of odd order groups.In this paper,we focus on the structure of finite groups whose maximal subgroups of even order satisfy some conditions.Let G be a finite group.If for every g∈G,we have x∈<H,Hg>,we called subgroup H is abnormal in G.If H permutes with each Sylow subgroup of G,we called subgroup H is S-quasinormal in G.This paper is divided into four chapters:In the first chapter,we introduce the research background of this paper and the previous research results.In the second chapter,we introduce some basic concepts and basic lemmas.And the results are as follows:Theorem 3.1:Let G be a non-SB-group of even order.If all maximal subgroups of G of even order are SB-groups,then G is solvable and |π(G)|≤3.Furthermore,one of the following statements holds.(1)G is a minimal non-SB-group;(2)|G|2=2,that is the order of the Sylow 2-subgroup T of G is 2,and one of the following holds:(2.1)G=T × P is a nilpotent group,where P ∈ Sylp(G);(2.2)G=T × P,and P ∈ Sylp(G)is an abelian group;(2.3)G=T × PQ and PQ is a minimal non-SB-group,here P and Q are Sylow p-subgroup and Sylow q-subgroup of G,respectively;(2.4)G=T(?)PQ,where[T,P]=1,CQ(T)=1 and Q is abelian,here P and Q are Sylow p-subgroup and Sylow q-subgroup of G,respectively.Theorem 3.2:Let G be a group.If all second maximal subgroups of G of even order are SB-groups,then G is solvable or G is isomorphic to one of the following groups:(1)PSL(2,2p),where 2p-1 is a prime;(2)PSL(2,q),where q≡3 or 5(mod 8);(3)SL(2,q),where q≡3 or 5(mod 8).The four chapter is summary and prospect.