{
  "abstract": "Let $G$ be a finite group and let $π(G)$ be the set of prime divisors of $|G|$. For each $p\\in π(G)$, choose a Sylow $p$-subgroup $G_p$ of $G$. The subgroup $Ψ(G)= \\prod_{p\\in π(G)} Φ(G_p)_G$ is called the \\textit{local Frattini subgroup} of $G$. Here, for $J \\leqslant G$, the symbol $J_G$ denotes the core of $J$ in $G$. The subgroup $Ψ(G)$ is a characteristic subgroup of $G$ that is contained in $Φ(G)$, the Frattini subgroup of $G$. We prove that $Ψ(G)$ satisfies many properties similar to $Φ(G)$. Let $J$ be a normal $p$-subgroup of $G$ with $J \\leqslant Φ(G)$, and let $S$ be a Sylow $p$-subgroup of $G$. John S. Rose proved that if $J \\leqslant \\mathbf{Z}(S)$, then $J \\leqslant Φ(S)$. So, in fact, $J \\leqslant Φ(S)_G \\leqslant Ψ(G)$. Kaplan, Lev, Mann, and Solomon found various other conditions that would ensure $J$ is contained in $Φ(S)$. In this article, we add a few more conditions to this list.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.16821",
  "author_total": 1,
  "authors": [
    {
      "name": "David G. Costanzo"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/a7rxma5fkgfvq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.16821",
  "fulltext_url": null,
  "id": "a7rxma5fkgfvq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.16821",
  "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.GR"
    ],
    [
      "openalex",
      "2607|Finite Group Theory Research"
    ]
  ],
  "openalex": "W7221811103",
  "pdf_url": "https://arxiv.org/pdf/2610.16821",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/a7rxma5fkgfvq.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:16Z",
      "native_id": "2610.16821",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.16821"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "On the local Frattini subgroup of a finite group",
  "title_text": "On the local Frattini subgroup of a finite group",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:16Z",
  "url": "https://research.draconitic.com/paper/a7rxma5fkgfvq",
  "venue": "arXiv",
  "versions": []
}