The Group Permanent Determines the Finite Abelian Group

Let $G$ be a finite abelian group of order $n$ and $\mathcal M_G$ the Cayley table of $G$. Let $\mathcal P(G)$ be the number of formally different monomials occurring in $\mathsf {per}(\mathcal M_G)$, the permanent of $\mathcal M_G$. In this paper, for any finite abelian groups $G$ and $H$, we prove the following characterization $$\mathcal P(G)=\mathcal P(H)\ \Leftrightarrow\ G\cong H.$$ It follows that the group permanent determines the finite abelian group, which partially answers an open question of Donovan, Johnson and Wanless. In fact, $\mathcal P(G)$ is closely related to zero-sum sequences over finite abelian groups and we shall prove the above characterization by studying a reciprocity of zero-sum sequences over finite abelian groups. As an application of our method, we show that $\mathcal P(G)>\mathcal P(C_n)$ for any non-cyclic abelian group $G$ of order $n$ and thereby answer an open problem of Panyushev.