Finite Abelian groups with positive genus subgroup intersection graphs

Abstract

The intersection graph of subgroups of a finite group [Formula: see text] is a graph whose vertices are all nontrivial subgroups of [Formula: see text] and in which two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text]. The non-orientable genus of a graph [Formula: see text] is the smallest positive integer [Formula: see text] such that [Formula: see text] can be embedded on [Formula: see text] ([Formula: see text]), where [Formula: see text] and [Formula: see text] are the surface obtained from the sphere by attaching [Formula: see text] handles and the sphere with [Formula: see text] added crosscaps, respectively. In this paper, we classify all finite abelian groups whose non-oritentable genus of intersection graphs of subgroups are 1–3, respectively.