The Influence Of The Set Of Orders Of Maximal Abelian Subgroups And The Greatest Common Divisor Of The Degree And Codegree Of An Irreducible Character Of A Finite Group On The Structure Of Finite Groups

We study two research topics in this thesis.The first topic is to study how the set of orders of maximal abelian subgroups influences the structure of a finite groups.Let G be a finite group,πmas(G)denotes the set of orders of maximal abelian subgroups of G.We prove the following theorems:Theorem 3.1 Let G be a finite group.If πmas(G)=πmas(S5),then G≌S5.Theorem 3.2 Let G be a finite group.If πmas(G)=πmas(S6),then G≌S6.Theorem 3.3 Let G be a finite group.If πmas(G)=πmas(S7),then G≌S7.Theorem 3.4 Let G be a finite group.If πmas(G)=πmas(S8),then G≌S8.Theorem 3.5 Let G be a finite group.If πmas(G)=πmas(S11),then G≌S11.Theorem 3.6 Let G be a finite group.If πmas(G)=πmas(Aut(M12)),then G≌Aut(M12).Theorem 3.7 Let G be a finite group.If πmas(G)=πmas(Aut(M22)),then G≌Aut(M22).The second topic is to study how the greatest common divisor of the irreducible character degree and corresponding codegree influences the structure of a finite group,and obtain two conclusions:Theorem 4.1 There does not exist any finite group G such that(χ(1),χc(1))is a prime for each χ∈Irr(G)#,where Irr(G)#is the set of non-principal irreducible characters of G.Theorem 4.2 Let G be a finite group.If gcd(χ(1),χc(1))is a prime for each χ∈Irr(G)\Lin(G),then G is solvable,where Lin(G)is the set of all linear irreducible characters of G.This thesis contains four chapters.In Chapter 1,the background and the main results.In Chapter 2,preliminaries.In Chapter 3,it is studied how the set of orders of maximal abelian subgroups influences the structure of a finite group.In Chapter 4,it is studied how the greatest common divisor of the irreducible character degree and corresponding codegree influences the structure of a finite group.