A systematic approach to group properties using its geometric structure

Abstract

The algebraic properties of a group can be explored through the relationship among its elements. In this paper, we define the graph that establishes a systematic relationship among the group elements. Let G be a finite group, the order product prime graph of a group G, is a graph having the elements of G as its vertices and two vertices are adjacent if and only if the product of their order is a prime power. We give the general presentation for the graph on dihedral groups and cyclic groups and classify finite dihedral groups and cyclic groups in terms of the order product prime graph as one of connected, complete, regular and planar. We also obtained some invariants of the graph such as its diameter, girth, independent number and the clique number. Furthermore, we used the vertex-cut of the graph in determining the nilpotency status of dihedral group. The graph on dihedral group is proven to be regular and complete only if the degree of the corresponding group is even prime power and connected for all prime power degree. It is also proven on cyclic group to be both regular, complete and connected if the group has prime power order. Additionally, the result turn out to show that any dihedral group whose order product prime graph's vertex-cut is greater than one is nilpotent. We also show that the order product prime graph is planar only when the degree of the group is three for dihedral group and less than five for cyclic group. Our final result shows that the order product prime graphs of any two isomorphic groups are isomophic.