On the Prime Graphs of Finite and Dihedral Groups

Abstract

Let G be a finite group. The order of the group G, denoted |G| is the number of elements in G. For any element g∈G, the order of g denoted o(g) is the smallest positive integer n ∈ N such that gⁿ = e. A graph of a finite group is said to be a prime graph Γ(G) if two vertices p,q are primes that divide |G|, with the vertices p and q connected by an edge if and only if the product pq divides o(g), the order of a group element. In this research, we study the prime graph of finite groups by focusing on one of these finite groups, i.e. the dihedral group with n sides which has order 2n. Next, we define and study the prime graph generated from 2 prime graphs of dihedral group Γ(D_m)∗Γ(D_n) and obtain the form of the prime graph of dihedral group which is forms the graph of K_n.

Keywords: Finite group, Dihedral group, Prime graph, Order of the group, Order of the group element.

2010 Mathematics Subject Classification. 26A25; 26A35