Discussion reveals the planarity of symmetric groups and groups of g-genus 1, indicating their structural properties.
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.
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Hamilton et al. (2026) studied this question.
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