The Influence Of The Arithmetic Conditions Of The Non-subnormal Subgroups On The Structure Of Finite Groups

For a long time,people used the arithmetic conditions of subgroup,which has always been one of the most important subject of finite group theory,to study the structure of finite group.The number of conjugacy classes of certain subgroups plays an so important role in the study of finite group structure that people from the different perspective are trying to expand its research scope.Many famous experts and scholars of group theory devoted themselves to the research,and they has gained a large number of research results.Inspired by these predecessors,in this paper,we mainly apply the idea of classification analysis and minimal counterexample.We consider the number of conjugacy classes and the isomorphism classes of the non-subnormal subgroup,and the number of conjugacy classes of non-subnormal non-nilpotent proper subgroup,to study the structure of finite group and its related properties.Let G be a finite group and let 1(G)denote the number of isomorphism classes of non-subnormal subgroups of G,lp(G)denote the number of isomorphism classes of non-subnormal non-nilpotent subgroups of G.Theorem 2.1.1 Let G be a finite group.If 1(G)≤|π(G)| + 3,then G is solvable.Theorem 2.1.2 Let G be a finite non-solvable group.Then l(G)= |π(G)| + 4 if and only if G(?)A5.Theorem 2.2.1 Let G be a finite non-solvable group.Then lp(G)≥ |π(G)|.In partic-ular,lp(G)= |π(G)| if and only G(?)A5 or SL(2,5).Theorem 2.2.2 Let G be a finite group.If G has at most 23 non-subnormal non-nilpotent proper subgroups,then G is solvable,except for G(?)A5 or SL(2,5).Theorem 2.2.3 Let G be a finite group.If G has at most 3 conjugacy classes of non-subnormal non-nilpotent proper subgroups,then G is solvable,except for G(?)A5 or SL(2,5).This thesis is divided into three chapters:in the first chapter we introduce some related concepts,the known conclusions and main lemmas.In the second chapter we study the influence of the non-subnormal subgroup and non-subnormal non-nilpotent proper subgroups on the structure of finite groups.In the third chapter we give a summary and some further research problems.