Difference graphs of finite abelian groups with two Sylow subgroups

The power graph and the enhanced power graph of a group G are simple graphs with vertex set G; two elements of G are adjacent in the power graph if one of them is a power of the other, and they are adjacent in the enhanced power graph if they generate a cyclic subgroup. The difference graph of a group G, denoted by D (G), is the difference of the enhanced power graph and the power graph of group G with all the isolated vertices removed. In this paper, we prove that, if a pair of finite abelian groups of order divisible by at most two primes have isomorphic difference graphs, then they are isomorphic.