{
  "abstract": "We extend Burnside's $p^a$-Theorem on conjugacy classes of prime-power size by proving that no non-abelian finite simple group possesses a conjugacy class of size $2p^n$ for any prime $p$ and $n\\geq 0$. Based on this, we partially extend a well-known result of L. Kazarin by showing that, given a finite group $G$, the subgroup generated by a conjugacy class $x^G$ of size $2p^n$ where $x$ has odd order, is a solvable normal subgroup of $G$. If instead the subgroup $\\langle x^G \\rangle$ generated by such a class $x^G$ is non-solvable (which forces $x$ to have even order), then $p= 5$ and $\\langle x^G \\rangle/S(\\langle x^G \\rangle)$ is isomorphic to the symmetric group $S_5$, where $S(\\langle x^G \\rangle)$ denotes the solvable radical of $\\langle x^G \\rangle$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.18193",
  "author_total": 2,
  "authors": [
    {
      "name": "Antonio Beltran"
    },
    {
      "name": "Changguo shao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fkmh678hnkxhd.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.18193",
  "fulltext_url": null,
  "id": "fkmh678hnkxhd",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.18193",
  "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",
      "2607|Finite Group Theory Research"
    ]
  ],
  "openalex": "W7221803146",
  "pdf_url": "https://arxiv.org/pdf/2610.18193",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-13",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fkmh678hnkxhd.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:20Z",
      "native_id": "2610.18193",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.18193"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Finite groups with conjugacy classes of size $2p^n$",
  "title_text": "Finite groups with conjugacy classes of size 2p^n",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:20Z",
  "url": "https://research.draconitic.com/paper/fkmh678hnkxhd",
  "venue": "arXiv",
  "versions": []
}