Hyperplanes in abelian groups and twisted signatures

We investigate the following question: if $A$ and $A^{\mathrm{\prime }}$ are products of finite cyclic groups, when does there exist an isomorphism $f:A\to A^{\mathrm{\prime }}$ which preserves the union of coordinate hyperplanes (equivalently, so that $f(x)$ has some coordinate zero if and only if $x$ has some coordinate zero)? We show that if such an isomorphism exists, then $A$ and $A^{\mathrm{\prime }}$ have the same cyclic factors; if all cyclic factors have order larger than $2$, the map $f$ is diagonal up to permutation, hence sends coordinate hyperplanes to coordinate hyperplanes. Thus one can recover the coordinate hyperplanes from knowledge of their union. This result is well-adapted for application to invariants with a certain multiplicativity property. As a model application, we show using twisted signatures that there exists a family of compact 4-manifolds $X(n)$ with $H_{1}X(n)=\mathbb{Z}\mathrm{/}n$ with the property that $\prod X(n_i)\cong \prod X(n_j^{\mathrm{\prime }})$ if and only if the factors may be identified (up to permutation), and that the induced map on first homology is (up to permutation) represented by a diagonal matrix.