A reciprocity on finite abelian groups involving zero-sum sequences

In this paper, we present a reciprocity on finite abelian groups involving zero-sum sequences. Let $G$ and $H$ be finite abelian groups with $(\mathrm{\mid }G\mathrm{\mid },\mathrm{\mid }H\mathrm{\mid })=1$. For any positive integer $m$, let $\mathsf{M}(G,m)$ denote the set of all zero-sum sequences over $G$ of length $m$. We have the following reciprocity $$\mathrm{\mid }\mathsf{M}(G,\mathrm{\mid }H\mathrm{\mid })\mathrm{\mid }=\mathrm{\mid }\mathsf{M}(H,\mathrm{\mid }G\mathrm{\mid })\mathrm{\mid }\mathrm{.}$$ Moreover, we provide a combinatorial interpretation of the above reciprocity using ideas from rational Catalan combinatorics. We also present and explain some other symmetric relationships on finite abelian groups with methods from invariant theory. Among others, we partially answer a question proposed by Panyushev in a generalized version.