{
  "id": "2001.06508",
  "version": "v1",
  "published": "2020-01-17T19:50:09.000Z",
  "updated": "2020-01-17T19:50:09.000Z",
  "title": "Compact groups with many elements of bounded order",
  "authors": [
    "Meisam Soleimani Malekan",
    "Alireza Abdollahi",
    "Mahdi Ebrahimi"
  ],
  "categories": [
    "math.GR",
    "math.GN"
  ],
  "abstract": "L\\'evai and Pyber proposed the following as a conjecture: Let $G$ be a profinite group such that the set of solutions of the equation $x^n=1$ has positive Haar measure. Then $G$ has an open subgroup $H$ and an element $t$ such that all elements of the coset $tH$ have order dividing $n$ (see Problem 14.53 of [The Kourovka Notebook, No. 19, 2019]). The validity of the conjecture has been proved in [Arch. Math. (Basel) 75 (2000) 1-7] for $n=2$. Here we study the conjecture for compact groups $G$ which are not necessarily profinite and $n=3$; we show that in the latter case the group $G$ contains an open normal $2$-Engel subgroup.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-01-17T19:50:09.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20E18",
      "20P05"
    ],
    "keywords": [
      "compact groups",
      "bounded order",
      "conjecture",
      "profinite group",
      "open normal"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}