Let A be a finite abelian group, and let U(A) be the group of units of \({\mathbb{Z}}A\) modulo torsion. Consider the maps \[ \prod_{C}U(C)\to^{\alpha}U(A)\to^{\beta}\prod_{K}U(K) \] where C and K run over the sets of cyclic subgroups and factor-groups of A, respectively. Here \(\alpha\) comes from a product of inclusions \(C\to A\) and \(\beta\) from the Wedderburn decomposition of \({\mathbb{Q}}A\). The main result of this paper asserts that if A is elementary abelian of order \(p^{n+1}\), then the cokernel of \(\beta\) \(\circ \alpha\) has order \(p^ m\), where \(m=(1/2)nr(1+...+p^ n)\) and r is the rank of U(C).