  Let $G$ be a finite abelian group. For any positive integers $d$ and $m$, let
$\varphi_G(d)$ 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 $\varphi_G(d)=\varphi_H(d)$ for any $d|(|G|,|H|)$.
We also consider an extension of this result to non-abelian groups in terms of
invariant theory.
