The Structures Of Some Finite Groups Under The Constraint Conditions Of Element Order Prime Graphs

It’ s well known that finite group theory is an important part of group theory.Researchers use element order and group order to study the structures of finite groups,and this article uses more intuitive element order prime graphs to study the structures of finite groups.The element-order prime graph is defined as:The prime factors of |G| are the vertexes of prime graph,it contains pq order elements,the two vertices p,q are connected by an edge.First,introducing the relevant knowledge of finite groups and element-order prime graphs,as well as the structures of finite groups under the restrict conditions of element-order prime graphs.Then,the structures of finite groups have been classified,discussed and researched from the number of isolated points、the number of vertices and the number of connected components of prime graph and so on.Based on the exploration of the coprimeness of the element orders and the properties of the group structures,combined with the graphic characteristics of the element orders graphs,the group structures of the finite group satisfying properties Q₅ is reserched.As follows:If G has the property Q₅,then|π（o（g））|<3,for each g ∈ G.If G has property Q₅,and t（G）=2,then |π（G）| ≤6.