Classification Of Prime Graphs Not Containing Triangles Of Finite Groups With Few Vertices

Group is a basic system in abstract algebra.Group theory have wide applications in mathematics and many modern science technologies,such as in theoretical physics,quantum mechanics,quantum chemistry,crystallography and cryptography and so on.The finite group is the basic part of the group theory.Describing the structures a-nd properties of finite groups by group orders and element orders is of the hot issue i-n recent years.many researchers used the two conditions of groups orders and the ele ments orders to characterize the finite group structures and have achieved some impor tant results.Because the structures of finite groups are more abstract and the graphs ar e more concrete,we study the structures of finite groups by using graphs to can make the group theory more concrete.In this thesis,the structures and properties of finite gr oups are described by the prime graphs.The definition of finite group prime graph is a s follow:?1?The prime factors of group order are the vertexes of prime graph;?2?The two vertices p and q are connected if and only if the group contains elements of order is pq.According as the prime graph contains triangle or not,the prime graph is divided into two categories.In this thesis,the corresponding group examples of the triangle-free prime graph with at most 6 vertices are studied to determine the finite group structure and properties corresponding to the prime graphs.First of all,all non-isomorphic graphs with not contain triangles of at most six vertices are totaling 65.According to the definition and composition conditions of the prime graph,there are five graphs are not prime graphs.In the rest of the graphs,we determined the group examples and related group properties corresponding to the 31 graphs,and obtained some results.1.If the prime graph?38??G?has no edge,thenp?G??4.whenp?G?=1,2,it is solv ble,whep?G?=3,4,it is not solvable.2.Let G be a finite group,the prime graph?38??G?is 3-regular if and only if it is a co mplete graph of four vertices.3.Let?38??G?be a prime graph of G withp?G?=6,If it does not contain triangle,G is nonsolved.For the simple groups of the prime graphs,we can get the following conclusions:Let G be a sporadic simple group or alternative simple group withp?G?=6,if?38??G?does not contain triangle,thenG?31?M₂₃.