Abelian groups with isomorphic intersection graphs (Q345967)

The intersection graph \(\mathcal{G}(G)\) of a group \(G\) has as vertices the proper non-trivial subgroups of \(G\) with an edge between the vertices \(X\) and \(Y\) if and only if \(X \cap Y\) is non-trivial. A natural question to ask is whether for some class of groups, \(\mathcal{G}(G)\) isomorphic to \(\mathcal{G}(H)\) as graphs implies \(G\simeq H\) as groups. In this paper, the authors show by elementary methods that this property holds for finite abelian groups with no non-trivial cyclic Sylow subgroup. A simple counterexample which shows that the latter condition is necessary is \(G = C_{p^n}\) and \(H = C_{q^n}\) for distinct primes \(p\) and \(q\) and \(n > 1\), which both have as intersection graph the complete graph on \(n-1\) vertices.