{
  "abstract": "For a finite group $G$ let $ω(G)$ denote its spectrum, the set of orders of its elements, and set $η(G)=|ω(G)|$, the number of distinct element orders. The set $ω(G)$ has been studied intensively, but its cardinality has received little systematic attention as an invariant in its own right. We develop the theory of $η$. After describing its behaviour under subgroups, quotients, sections, direct products and Frobenius extensions, we prove the two-sided bound $1+\\sum_{p\\mid|G|}v_p(\\exp G)\\leη(G)\\leτ(\\exp G)$, in which the lower equality characterises the CP-groups, those all of whose elements have prime-power order, and the upper equality the groups whose spectrum realises every divisor of the exponent; nilpotent groups lie at the upper end, giving $η(G)=τ(\\exp G)$. We also prove $η(G)\\leτ(|G|)$ with equality only for cyclic groups, and $η(G)\\geπ(|G|)+1$ with equality only when every nontrivial element has prime order. We show that $η(G)\\le3$ forces solvability by an argument resting only on Burnside's theorem, and, combining this with the recognition of $A_5$ by its spectrum, that $A_5$ is the unique non-solvable group with $η(G)=4$. For the symmetric and alternating groups we record self-contained proofs of closed criteria for membership in the spectrum through a single arithmetic function, placing the associated counting sequences in the present framework, and we determine the extreme values of $η$ over all groups of a fixed order.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.13016",
  "author_total": 2,
  "authors": [
    {
      "name": "Manoj Kumar Singh"
    },
    {
      "name": "Sumant Kumar"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fr4p7vvay5ka1.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.13016",
  "fulltext_url": null,
  "id": "fr4p7vvay5ka1",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.13016",
  "language": null,
  "licence": "CC-BY-SA-4.0",
  "licence_label": "CC BY-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.GR"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2609.13016",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-11",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fr4p7vvay5ka1.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:11Z",
      "native_id": "2609.13016",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.13016"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "The number of distinct element orders in a finite group",
  "title_text": "The number of distinct element orders in a finite group",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:11Z",
  "url": "https://research.draconitic.com/paper/fr4p7vvay5ka1",
  "venue": "arXiv",
  "versions": []
}