We investigate the following question: If $A$ and $A’$ are products of finite cyclic groups, is there an isomorphism $f: A \to A’$ that holds the union of coordinate hyperplanes? When? f(x)$ has coordinate zero iff $x$ has coordinate zero)?
If such an isomorphism exists, we show that $A$ and $A’$ have the same circular factor. If the order of all cyclic factors is greater than $2$, then the map $f$ is diagonal up to the permutation, so it sends the coordinate hyperplane to the coordinate hyperplane. Therefore, we can recover the coordinate hyperplane from the knowledge of the union.
This result is suitable for application to certain multiplicative invariants. As an application of the model, the $\prod X(n_i ) \cong \prod X(n’_j)$ factors are identified (up to permutation) and the induced map on the first homology is (up to permutation) the diagonal matrix only if indicated.
