Finite Abelian groups with positive genus subgroup intersection graphs

The intersection graph of subgroups of a finite group G is a graph whose vertices are all nontrivial subgroups of G and in which two distinct vertices H and K are adjacent if and only if H n K ? 1. The non-orientable genus of a graph G is the smallest positive integer n such that G can be embedded on S-n (N-n), where S(n )and N(n )are the surface obtained from the sphere by attaching n handles and the sphere with n 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.