Orientable and non-orientable genus of the intersection power graph of finite groups

The intersection power graph ΓI (G) of a group G is defined as follows: take G as vertex set and two distinct vertices x and y are adjacent in ΓI (G) if there exists a non-identity element z ∈ G such that xm=z=yn, for some m, n ∈ N and e is adjacent to all other vertices, where e is the identity element of the group G. In this paper, certain finite groups whose intersection power graphs can be embedded on the torus or projective plane are classified.