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

TL;DR

This paper establishes a symmetry relation between zero-sum sequences in finite abelian groups based on element order distributions, and explores extensions to non-abelian groups using invariant theory.

Contribution

It proves a reciprocity relation for zero-sum sequences involving group order and element order counts, extending to non-abelian groups.

Findings

Equality of zero-sum sequence counts implies identical element order distributions.

The reciprocity holds if and only if the groups have matching element order counts.

Extension to non-abelian groups via invariant theory is discussed.

Abstract

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.