{
  "abstract": "Let $G$ be a finite abelian group of order $n$ and $X$ a finite $G$-set. We give a necessary and sufficient condition on $G$ for any $X$ to be determined up to isomorphism by certain combinatorial data. Specifically, if $M_H^X$ is the number of $H$-orbits in $X$, and $a_d^X$ is the number of $G$-orbits in $X$ of cardinality $d$, we characterize when equalities $M_H^{X_1} = M_H^{X_2}$ and $a_d^{X_1}=a_d^{X_2}$ for all $H \\leq G$ and $d|n$, respectively, force an isomorphism $X_1 \\simeq X_2$. Our main result is that this occurs if and only if $G$ is cyclic or $G$ is the direct sum of a finite cyclic group and a cyclic group of prime order. We conclude with an arithmetic application to $L$-function factorization.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.16880",
  "author_total": 5,
  "authors": [
    {
      "name": "Carson Gull"
    },
    {
      "name": "Arthur Illidge"
    },
    {
      "name": "Benjamin Mudrak"
    },
    {
      "name": "Joshua Park"
    },
    {
      "name": "Shaver Phagan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/69kgx82trajb1.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.16880",
  "fulltext_url": null,
  "id": "69kgx82trajb1",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.16880",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.GR"
    ],
    [
      "openalex",
      "2607|Finite Group Theory Research"
    ]
  ],
  "openalex": "W7221821111",
  "pdf_url": "https://arxiv.org/pdf/2610.16880",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/69kgx82trajb1.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:16Z",
      "native_id": "2610.16880",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.16880"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "On a Combinatorial Characterization of Abelian Group Actions",
  "title_text": "On a Combinatorial Characterization of Abelian Group Actions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:16Z",
  "url": "https://research.draconitic.com/paper/69kgx82trajb1",
  "venue": "arXiv",
  "versions": []
}