{
  "id": "2201.12821",
  "version": "v1",
  "published": "2022-01-30T14:06:02.000Z",
  "updated": "2022-01-30T14:06:02.000Z",
  "title": "A reciprocity on finite abelian groups involving zero-sum sequences II",
  "authors": [
    "Mao-Sheng Li",
    "Hanbin Zhang"
  ],
  "comment": "16 pages",
  "categories": [
    "math.CO",
    "math.NT"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-01-30T14:06:02.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "finite abelian group",
      "zero-sum sequences",
      "reciprocity",
      "non-abelian groups",
      "invariant theory"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}