{
  "abstract": "A classic problem in additive number theory asks whether a given subset of integers has more sums or differences. This inquiry naturally extends to finite groups, where one compares the sizes of the product set $AA$ and the quotient set $AA^{-1}$ for a subset $A \\subseteq G$. A finite group is considered product-dominant if it contains a higher proportion of subsets where $|AA| > |AA^{-1}|$, and quotient-dominant if the reverse is true. We establish statistical criteria to determine the dominance behavior of large finite groups by analyzing the moments of the random variable $d(A) := |AA| - |AA^{-1}|$. By translating product and quotient relationships into graph-theoretic constructions, we compute these moments through counting independent sets. Our main result demonstrates that for a growing sequence of finite groups $G_k$ with center size $Z_k$ and involution count $T_k$, the condition $Z_k = o(T_k)$ implies the groups are eventually product-dominant. Conversely, if $T_k \\cdot \\sqrt{3}^{R_k/Z_k} = o(Z_k)$ (where $R_k$ denotes the number of elements whose square is central), the groups are eventually quotient-dominant. Using these asymptotic conditions, we resolve the Miller--Vissuet conjecture by proving that all dihedral groups $D_n$ for $n \\geq 3$ are product-dominant. Furthermore, we classify the dominance behavior of several major families of groups, proving that symmetric groups and non-abelian finite simple groups are product-dominant, while groups of odd order and certain abelian groups are quotient-dominant. Finally, for groups that fall outside our asymptotic conditions, we derive an explicit formula for the first moment utilizing the irreducible characters and conjugacy classes of the group.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.12912",
  "author_total": 8,
  "authors": [
    {
      "name": "Yuval Amit"
    },
    {
      "name": "Lucas Chen"
    },
    {
      "name": "Christopher Housholder"
    },
    {
      "name": "Joshua Im"
    },
    {
      "name": "Joshua Khan"
    },
    {
      "name": "Steven J. Miller"
    },
    {
      "name": "Mateo Palomares"
    },
    {
      "name": "Devayani Pradhan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0x7n2xyf9b6c2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.12912",
  "fulltext_url": null,
  "id": "0x7n2xyf9b6c2",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.12912",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2607|Finite Group Theory Research"
    ]
  ],
  "openalex": "W7221542030",
  "pdf_url": "https://arxiv.org/pdf/2610.12912",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0x7n2xyf9b6c2.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:28:40Z",
      "native_id": "2610.12912",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.12912"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Product and Quotient Set Dominance in Finite Groups",
  "title_text": "Product and Quotient Set Dominance in Finite Groups",
  "type": "preprint",
  "updated_at": "2026-10-30T12:28:40Z",
  "url": "https://research.draconitic.com/paper/0x7n2xyf9b6c2",
  "venue": "arXiv",
  "versions": []
}