Some Sufficient Conditions For A Finite Group To Be Solvable

For a long time,the relationship between the properties of subgroups and the structure of finite groups is one of the most important research topics in group theory.In this paper we study the influence of the number of supersoluble subgroups and the number of conjugate classes of supersoluble subgroups,and the automizers of subgroups on the solvability of finite groups.In the first chapter,we introduce the research background.In the second chapter,we introduce some basic concepts and lemmas.The third chapter is divided into two parts:In chapter 3.1,we obtain some sufficient conditions for solvability and related structures by studying the influence of the number of supersoluble subgroups and the number of the conjugate classes of supersoluble subgroups.In this process,we focus on the counterexample of minimal order and the structure of minimal simple groups.The results are as follows:Theorem 3.1.1 Let G be a finite group.We denote by ?(G)the number of supersoluble subgroups of G.(1)If ?(G)<53,then G is solvable.(2)G is a non-solvable group with ?(G)= 53 if and only if G(?)A5.Theorem 3.1.2 Let G be a finite group.We denote by ?(G)the number of conjugate classes of supersoluble subgroups.(1)If ?(G)<7,then G is solvable.(2)G is a non-solvable group with ?(G)= 7 if and only if G(?)A5.In chapter 3.2,we discuss the influence of small automizers for Sylow subgroups on the structure of finite groups,and obtain a sufficient condition for solvability of finite groups.We solve some of the problems raised by Deaconescu and Walls that describe the structure of small automizers for Sylow subgroups.The results are as follows:Theorem 3.2.1 Let G be a solvable group with small automizers for Sylow subgroups.Then G is nilpotent.Theorem 3.2.2 Let G be a finite group with small automizers for Sylow subgroups.Then G is solvable.