{
  "abstract": "Given two (closed) subgroups H, K of a profinite group G, we write Pr(H,K) for the probability that a random pair from HxK commutes. Here we are concerned with profinite groups containing Sylow subgroups P, Q such that Pr(P,Q) > 0. First, we show that if G is a profinite group containing a Sylow 2-subgroup L and a Sylow p-subgroup P , where p is odd, such that Pr(L,P) is positive, then G is virtually pro-p-soluble (Theorem 1.1). Then we handle similar issues for profinite groups admitting coprime automorphisms. In particular, we prove that if G is a profinite group admitting a group of coprime automorphisms A such that there is a Sylow 2-subgroup L of C_G(A) and an A-invariant Sylow p-subgroup P of G, where p is odd, for which Pr(L,P) > 0, then G is virtually pro-p-soluble (Theorem 1.3). On the other hand, if we only have Pr([L,A],[P,A]) > 0, then [G,A] need not be virtually pro-p-soluble. We show that in this case [G,A] has an open normal subgroup of non-p-soluble length at most 1 (Theorem 1.4). Furthermore, if P is an A-invariant Sylow p-subgroup of G such that Pr([P, A],[P, A]^x) > 0 for every x in G, then [G,A] has an open normal subgroup of non-p-soluble length at most 1 (Theorem 1.5).",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.29925",
  "author_total": 4,
  "authors": [
    {
      "name": "Eloisa Detomi"
    },
    {
      "name": "Robert M. Guralnick"
    },
    {
      "name": "Marta Morigi"
    },
    {
      "name": "Pavel Shumyatsky"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2yyppecjyq1vv.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.29925",
  "fulltext_url": null,
  "id": "2yyppecjyq1vv",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.29925",
  "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": "W7214389098",
  "pdf_url": "https://arxiv.org/pdf/2609.29925",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2yyppecjyq1vv.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:15Z",
      "native_id": "2609.29925",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.29925"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Coprime automorphisms of profinite groups and the commuting probability of Sylow subgroups",
  "title_text": "Coprime automorphisms of profinite groups and the commuting probability of Sylow subgroups",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:15Z",
  "url": "https://research.draconitic.com/paper/2yyppecjyq1vv",
  "venue": "arXiv",
  "versions": []
}