Let A be a finite abelian group, let \(U^.(A)\) be the group of units of \({\mathbb{Z}}A\) modulo torsion and let \({\dot \alpha}\): \(\prod_{C}U^.(C)\to U^.(A)\) be the natural homomorphism, where the product is direct and C runs over all cyclic subgroups \(\neq 1\) of A. In this note the authors prove the following result. Theorem. If A is an elementary abelian p-group, where p is a regular prime, then \({\dot \alpha}\) is an isomorphism.