On The Connection Between The Commuting Probability And The Structure Of Finite Groups

Let G be a finite group. G is said to be a CLT-group, if G satisfies the converse of Lagrange’s Theorem. That is for each factor of |G|, G has at least one subgroup H of order n. Let d（G）=k（G）/|G| denote the commuting probability, where k（G） is the number of conjugacy classes of G.In the investigation of finite groups, using order of group or properties of subgroup or properties of element to protray the structure and discuss properties of finite groups is a main direction and a common approach. In this thesis, using order of G and probability of element of G commute, we explore properties of G, and we get some new characterizations of G is a CLT-group and G is a supersolvable group. In accordance with the contents divided into two chapters:The first chapter, we mainly analyze concepts of the CLT-group and the commuting probability and other related courses, introduce them investigative background, the basic definitions and some relevant known results, the main properties and correlative lemmas which are related to the CLT-group and the commuting probability. The second chapter, by using the commuting probability to research the structure of groups. The main results are as follows:Theorem 2.1.1 Let G be a finite group, if d（G）> yg, then:（1） G is a CLT-group;（2） G isoclinic A4;（3） G/Z（G） isoclinic A4;（4） G isoclinic （C₃ × C₃） × C₄;（5） G isoclinic （C₃ × C₃）× C₈;（6） G isoclinic （C₃ × C₃） × （C₄ × C2）;（7） G isoclinic （C2 × C2 × C2） × C14;（8） G/Z（G） isoclinic （C2 × C6） × C₃;（9） G isoclinic （C2 × C6） × C₃.Theorem 2.1.2 Let G be a finite group of odd order, if d（G）>29/225, then:（1） G is a CLT-group;（2） G isoclinic （C5 × C5） × C₃,（3） G isoclinic （C₃ × C₃ × C₃）× C13.Theorem 2.2.1 Let G be a finite group of odd order, if d（G）>29/225, then:（1） G is supersolvable;（2） G isoclinic （C5 × C5） × C₃;（3） G isoclinic （C₃ × C₃ × C₃） × C13;（4） G isoclinic （C₃ × G₃ × C₃） × G₃9.