For a group Formula: see text with generating set Formula: see text, the Formula: see text-graph of Formula: see text, denoted Formula: see text, is the graph whose vertices are distinct cosets of Formula: see text in Formula: see text. Two distinct vertices are joined by an edge when the set intersection of the cosets is nonempty. From 12, we know that finite Abelian groups that are cyclic or are isomorphic to Formula: see text or Formula: see text are Formula: see text-planar. In this discussion, we discuss the planarity of the symmetric group and discuss groups of Formula: see text-genus 1.
Hamilton et al. (Fri,) studied this question.