Finite Groups Whose Co-prime Order Graphs Have Positive Genus

Abstract

Given a finite group G, the co-prime order graph of G is the simple undirected graph whose vertex set is G, and two distinct vertices x; y are adjacent if gcd(o(x); o(y)) = 1 or p, where p is a prime, and o(x) and o(y) are the orders of x and y, respectively. In this paper, we prove that, for a fixed positive integer k, there are finitely many finite groups whose co-prime order graphs have (non)orientable genus k. As applications, we classify all finite groups whose co-prime order graphs have (non)orientable genus one and two.