The Noncommuting Graphs And Structures Of Finite Groups

Generally,there is an intimate relation between the groups and graphs,and in many occasions properties of graphs give rise to some properties of groups and vice versa.For instance,the prime graphΓ(G) associated with a finite group G introduced by Gruenberg and Kegel who give a classification of finite groups by the number of connected componetes of their prime graph(cf.[13]).After that,there are many articles which give the characterizations of simple groups from the different conditions such as the conditions "the order of group and the set of element orders " and "the set of element orders".One of these graphs that has attracted the attention of many authors is the non-commuting graph▽(G) associated with a finite group G.It has been defined in [2]as follows:the vertex set of▽(G) is G\Z(G) with two vertices x and y joined by an edge whenever the commutator of x and y is not the identity.In 2006,A.Abdollahi,S.Akbari and H.R.Maimani put forword the following conjectures(cf.[1]):Conjecture 1 Let G and H are finite non-abelian groups such that▽(G)▽(H),then|G|=|H|.Conjecture 2 Let G be a finite non-abelian simple groups and H a finite non-abelian group satisfying▽(G)≌▽(H),then G≌H.In recent years,many scholars study the finite simple group by its non-commuting graph.Many research articles get that two non-commutings of some simple groups are isomorphic if and only if the two simple groups are the same.But so far there is nobody to study the connection between the non-commuting graphs and structures of finite non-simple groups.This article take the structure of finite group into account. It study the the finite non-simple group by its non-commuting graph and get some conclusions.It consists of the three following chapters:In Chapter 1,we introduce some symbols and basic concepts that we usually use in the paper.Moreover,we introduce some backgrounds and results of our research.In Chapter 2,study the non-commuting graph and structure of some finite nonsimple groups by the classify theory of groups.In Chapter 3,study the non-commuting graph and structure of some finite nonsimple groups by elementary knowledge but classify theory of groups.The validity of conjecture 1 and conjecture 2 is known for some non-simple groups.