A reciprocity on finite abelian groups involving zero-sum sequences II

Abstract

Let $G$ be a finite abelian group. For any positive integers $d$ and $m$, let $\ddot{I}{†}_G(d)$ be the number of elements in $G$ of order $d$ and $\mathsf{M}(G,m)$ be the set of all zero-sum sequences of length $m$. In this paper, for any finite abelian group $H$, we prove that $$|\mathsf{M}(G,|H|)|=|\mathsf{M}(H,|G|)|$$ if and only if $\ddot{I}{†}_G(d)=\ddot{I}{†}_H(d)$ for any $d|(|G|,|H|)$. We also consider an extension of this result to non-abelian groups in terms of invariant theory.

16 pages