| It is important in the study of groups to find the relationship between the structure of finite groups and conjugacy classes. Over the last years this has been a flourishing and active area of research and many results have been achieved. The main problems discussed in this thesis is the relationship between the graph of conjugacy class lengths and the structure of finite groups. The graph of conjugacy class lengths T(G) has the vertex set Cl(G) and an edge between|Ci|,|Cj|∈Cl(G) if and only if(|Ci|,|Cj|)>1. The properties of the graph (the vertices and edges) can be used to classify the structure of finite groups.The graphs with4vertices have been discussed in this thesis and there are18graphs. According to the definition and properties of graph of conjugacy class lengths, graph has only one isolated vertex when G=P×H with one conjugacy class length. Graph only has two isolated vertices if and only if G is a quasi-Frobenius group with an abelian kernel and a complement.Conjugacy class lengths and structure of groups can be calculated by the GAP, and a classification of finite groups of order100or less is given in this thesis.The graph of conjugacy class lengths focus on all the conjugacy class lengths, but it is meaningful to study any one conjugacy class length. A geoup is called SCLD-Group if every square of conjugacy class lengths of elements of G divides the order|G|. In the fourth chapter of this thesis, some results are proved on SCLD-Groups:(1) Abelian groups must be SCLD-Groups;(2) If a finite group is a SCLD-Group, it must not be a simple group, almost simple group or Frobenius group;(3) The Sylow p-subgroups of nilpotent group are all SCLD-Groups if and only if nilpotent group is SCLD-Group. |
