{
  "id": "2409.11165",
  "version": "v1",
  "published": "2024-09-17T13:18:17.000Z",
  "updated": "2024-09-17T13:18:17.000Z",
  "title": "Commuting probability for the Sylow subgroups of a profinite group",
  "authors": [
    "Eloisa Detomi",
    "Marta Morigi",
    "Pavel Shumyatsky"
  ],
  "categories": [
    "math.GR"
  ],
  "abstract": "Given two subgroups $H,K$ of a compact group $G$, the probability that a random element of $H$ commutes with a random element of $K$ is denoted by $Pr(H,K)$. We show that if $G$ is a profinite group containing a Sylow $2$-subgroup $P$, a Sylow $3$-subgroup $Q_3$ and a Sylow $5$-subgroup $Q_5$ such that $Pr(P,Q_3)$ and $Pr(P,Q_5)$ are both positive, then $G$ is virtually prosoluble (Theorem 1.1). Furthermore, if $G$ is a prosoluble group in which for every subset $\\pi\\subseteq\\pi(G)$ there is a Hall $\\pi$-subgroup $H_\\pi$ and a Hall $\\pi'$-subgroup $H_{\\pi'}$ such that $Pr(H_\\pi,H_{\\pi'})>0$, then $G$ is virtually pronilpotent (Theorem 1.2).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-09-17T13:18:17.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "sylow subgroups",
      "commuting probability",
      "random element",
      "compact group",
      "profinite group containing"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}