{
  "id": "2208.04666",
  "version": "v1",
  "published": "2022-08-09T11:20:02.000Z",
  "updated": "2022-08-09T11:20:02.000Z",
  "title": "Nilpotent probability of compact groups",
  "authors": [
    "Alireza Abdollahi",
    "Meisam Soleimani Malekan"
  ],
  "categories": [
    "math.GR"
  ],
  "abstract": "Let $k$ be any positive integer and $G$ a compact (Hausdorff) group. Let $\\mf{np}_k(G)$ denote the probability that $k+1$ randomly chosen elements $x_1,\\dots,x_{k+1}$ satisfy $[x_1,x_2,\\dots,x_{k+1}]=1$. We study the following problem: If $\\mf{np}_k(G)>0$ then, does there exist an open nilpotent subgroup of class at most $k$? The answer is positive for profinite groups and we give a new proof. We also prove that the connected component $G^0$ of $G$ is abelian and there exists a closed normal nilpotent subgroup $N$ of class at most $k$ such that $G^0N$ is open in $G$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-08-09T11:20:02.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20E18",
      "20P05"
    ],
    "keywords": [
      "compact groups",
      "nilpotent probability",
      "open nilpotent subgroup",
      "closed normal nilpotent subgroup",
      "profinite groups"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}