On the genus of a graph related to the join of subgroups of finite abelian group

Abstract

Let G be a finite group which is not a cyclic p-group, p is a prime number. The undirected simple graph \(\varGamma (G)\) whose vertices are the proper subgroups of G which are not contained in the Frattini subgroup of G and two vertices \(H_1\) and \(H_2\) are joined by an edge if and only if \(G=\left\langle H_1,H_2 \right\rangle \). In this paper, we classify all finite abelian groups G for which \(\varGamma (G)\) has genus one.