The Study Of Graph And Product Of Conjugacy Classes Of Finite Groups

It is an important subject in the theory of finite groups to depicting the structures of finite groups from some arithmetic properties of their conjugacy classes,many group theory scholars at home and abroad about the research has received a number of research results.In this paper,by defining some graph associated with the conjugate classes of finite groups,study the nature of these graphs and the number of properties on the impact of group structure and the use of conjugate classes of some number of properties of the product to depict the structure of finite group,This paper is divided into four chapters,mainly has the following contents:The first chapter mainly introduces the paper work related literature background and main research content.The second chapter mainly lists some basic concepts,lemmas and proofs of the lemmas related to this thesis.In Chapter 3,we investigate how the number of some graph associated with the conjugate classes of a finite group influences its structure.Let G be a finite group and g be an element of G,denote by o(g)the order of g.Letg1G,g2G,…,gbG be n distinct noncentral conjugacy classes of G,if(o(g1),o(g2),…,O(g2))= 1,that is,the orders of g1,g2,…gn are setwise co-prime,then we say G has the property Pn.You xingzhong studied the structures of finite groups satisfying property P3 and P4,When G is non-solvable group,[61]didn't investigate the structures of finite groups with property P5.This article will use the simple group classification theorem depicting the structures of non-abelian groups satisfying property P5.Because with the increase of n,depicting satisfy property Pn group structure will be more complex,also is full of difficulties,this paper studies the structures of finite group when G is a specific group of classes(nilpotent group,Frobenious group and permutation group)and satisfy the properties of P6.In Chapter 4,we studied the product of the conjugate classes of finite group G to the influence of the structure of G.First,some properties of regular subset of finite group G,and the power of conjugate classes of properties were studied.For any xG ? con(G),xG = {x9|g ? G} ? x[x,G],[x,G]is the non-empty subset of G,The article do some research to[x,G],Considering the subgroup of G is[x,G]equivalent conditions and related issues.For any xG,yG ? con(G),The product xGyG of xG and yG is formal subset of G,So xGyG is set of some conjugate classes of G,kG(xGyG)for xGyG contains the number of conjugate classes of G,In this paper,we study the influence of kG(xGyG)= 1 on group structure under some conditions of xG,yG,at the same time G is abelian group if and only if kG(xGyG)= 1 for any x,y ? G.