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 graphs 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 thevertex-cut of the graph in determining the nilpotency status of dihedralgroups. The graph on dihedral groups 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 groups 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 groups and less than five for cyclic groups. Our final result shows that the order product prime graphs of any two isomorphic groups are isomophic.

Graph coloring using commuting order product prime graph

“…In this paper, we extend the work of Bello et al in [6], by introducing commuting order product prime graph of finite groups, as a graph having the elements of a group as its vertices, and two vertices x, y are adjacent if and only if O(x)O(y) = p s , s > 0 and xy = yx. We then investigate the general presentations of the graph on cyclic groups, dihedral groups, generalized quaternion groups and quasidihedral groups.…”

“…In addition, Chattopadhyay et al in [10] determined the exact value for the connectivity of the power graph on finite cyclic groups. Bello et al define order product prime graph of finite groups in [6], as a graph having the elements of a group as its vertices and any two vertices are adjacent if and only if the product of their order is a prime power. This graph started attracting the attention of researchers, for instance in [5], the topological indices of the graph are explored.…”