{
  "abstract": "Let $A$ be a nonempty subset of a group $G$. A right subfactor relative to $A$ is a subset $B$ maximal under inclusion among those for which every element of $AB$ has a unique representation as $ab$, with $a\\in A$ and $b\\in B$. Left subfactors are defined analogously using $BA$. The infima and suprema of the cardinalities of right and left subfactors define four \\emph{subindices}. A subset is called \\emph{index-stable} if its four subindices coincide. For any infinite group $G$, every nonempty finite subset of $G$ is index-stable. For finite groups, the situation is more complicated. We prove that for a two-element subset of a finite group, the left and right subindices coincide pairwise. Let $A=\\{x,y\\}$ with $x\\ne y$, and let $X$ be the cyclic subgroup generated by $x^{-1}y$. Then the lower subindex of $A$ is $|G:X|\\cdot\\lceil |X|/3\\rceil$, and the upper subindex is $|G:X|\\cdot\\lfloor |X|/2\\rfloor$. These formulas imply that $A$ is index-stable if and only if $|X|$ is one of $2,3,4,5,7$. For any two-element subset of a finite group $G$, the lower subindex is at least $\\lceil |G|/3\\rceil$, and the upper subindex is at most $\\lfloor |G|/2\\rfloor$. We establish necessary and sufficient conditions for equality in each of these bounds. All two-element subsets of a finite group are index-stable if and only if every element of the group has order $1$, $2$, $3$, $4$, $5$, or $7$. We give a complete structural classification of such groups.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.39463",
  "author_total": 2,
  "authors": [
    {
      "name": "M. H. Hooshmand"
    },
    {
      "name": "Mikhail Kabenyuk"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5avsdeweq6cz4.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.39463",
  "fulltext_url": null,
  "id": "5avsdeweq6cz4",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.39463",
  "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.GR"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.39463",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5avsdeweq6cz4.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.39463",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.39463"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Subindices of two-element subsets of finite groups",
  "title_text": "Subindices of two-element subsets of finite groups",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/5avsdeweq6cz4",
  "venue": "arXiv",
  "versions": []
}