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.
@article{10_37236_13332,
author = {Mao-sheng Li and Hanbin Zhang},
title = {The group permanent determines the finite abelian group},
journal = {The electronic journal of combinatorics},
year = {2024},
volume = {31},
number = {4},
doi = {10.37236/13332},
zbl = {1557.11006},
url = {http://geodesic.mathdoc.fr/articles/10.37236/13332/}
}
TY - JOUR
AU - Mao-sheng Li
AU - Hanbin Zhang
TI - The group permanent determines the finite abelian group
JO - The electronic journal of combinatorics
PY - 2024
VL - 31
IS - 4
UR - http://geodesic.mathdoc.fr/articles/10.37236/13332/
DO - 10.37236/13332
ID - 10_37236_13332
ER -
%0 Journal Article
%A Mao-sheng Li
%A Hanbin Zhang
%T The group permanent determines the finite abelian group
%J The electronic journal of combinatorics
%D 2024
%V 31
%N 4
%U http://geodesic.mathdoc.fr/articles/10.37236/13332/
%R 10.37236/13332
%F 10_37236_13332
Mao-sheng Li; Hanbin Zhang. The group permanent determines the finite abelian group. The electronic journal of combinatorics, Tome 31 (2024) no. 4. doi: 10.37236/13332
