{
  "id": "2112.08681",
  "version": "v2",
  "published": "2021-12-16T07:58:36.000Z",
  "updated": "2022-07-06T14:06:15.000Z",
  "title": "On the commuting probability of p-elements in a finite group",
  "authors": [
    "Timothy C. Burness",
    "Robert M. Guralnick",
    "Alexander Moretó",
    "Gabriel Navarro"
  ],
  "comment": "Revised according to referee's report. To appear in Algebra & Number Theory",
  "categories": [
    "math.GR"
  ],
  "abstract": "Let $G$ be a finite group, let $p$ be a prime and let ${\\rm Pr}_p(G)$ be the probability that two random $p$-elements of $G$ commute. In this paper we prove that ${\\rm Pr}_p(G) > (p^2+p-1)/p^3$ if and only if $G$ has a normal and abelian Sylow $p$-subgroup, which generalizes previous results on the widely studied commuting probability of a finite group. This bound is best possible in the sense that for each prime $p$ there are groups with ${\\rm Pr}_p(G) = (p^2+p-1)/p^3$ and we classify all such groups. Our proof is based on bounding the proportion of $p$-elements in $G$ that commute with a fixed $p$-element in $G \\setminus \\textbf{O}_p(G)$, which in turn relies on recent work of the first two authors on fixed point ratios for finite primitive permutation groups.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2022-07-06T14:06:15.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "finite group",
      "commuting probability",
      "p-elements",
      "finite primitive permutation groups",
      "abelian sylow"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}