{
  "id": "2307.09868",
  "version": "v1",
  "published": "2023-07-19T09:57:25.000Z",
  "updated": "2023-07-19T09:57:25.000Z",
  "title": "On the commuting probability of $π$-elements in finite groups",
  "authors": [
    "Juan Martínez"
  ],
  "comment": "16 pages",
  "categories": [
    "math.GR"
  ],
  "abstract": "Let $G$ be a finite group and let $\\pi$ be a set of primes. In this work, we prove that $G$ possesses a normal and abelian Hall $\\pi$-subgroup if and only if the probability that two random $\\pi$-elements of $G$ commute is larger than $\\frac{p^2+p-1}{p^3}$, where $p$ is the smallest prime in $\\pi$. We also prove that if $x$ is a $\\pi$-element not lying in $O_{\\pi}(G)$, then the proportion of $\\pi$-elements commuting with $x$ is at most $1/p$, where $p$ is the smallest prime in $\\pi$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-07-19T09:57:25.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20D20"
    ],
    "keywords": [
      "finite group",
      "commuting probability",
      "smallest prime",
      "abelian hall",
      "proportion"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}