{
  "id": "1807.06452",
  "version": "v1",
  "published": "2018-07-14T13:21:24.000Z",
  "updated": "2018-07-14T13:21:24.000Z",
  "title": "Compact groups all elements of which are almost right Engel",
  "authors": [
    "E. I. Khukhro",
    "P. Shumyatsky"
  ],
  "comment": "arXiv admin note: substantial text overlap with arXiv:1610.02079, arXiv:1512.06097",
  "categories": [
    "math.GR"
  ],
  "abstract": "We say that an element $g$ of a group $G$ is almost right Engel if there is a finite set ${\\mathscr R}(g)$ such that for every $x\\in G$ all sufficiently long commutators $[...[[g,x],x],\\dots ,x]$ belong to ${\\mathscr R}(g)$, that is, for every $x\\in G$ there is a positive integer $n(x,g)$ such that $[...[[g,x],x],\\dots ,x]\\in {\\mathscr R}(g)$ if $x$ is repeated at least $n(x,g)$ times. Thus, $g$ is a right Engel element precisely when we can choose ${\\mathscr R}(g)=\\{ 1\\}$. We prove that if all elements of a compact (Hausdorff) group $G$ are almost right Engel, then $G$ has a finite normal subgroup $N$ such that $G/N$ is locally nilpotent. If in addition there is a uniform bound $|{\\mathscr R}(g)|\\leq m$ for the orders of the corresponding sets, then the subgroup $N$ can be chosen of order bounded in terms of $m$. The proofs use the Wilson--Zelmanov theorem saying that Engel profinite groups are locally nilpotent and previous results of the authors about compact groups all elements of which are almost left Engel.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-07-14T13:21:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20D25",
      "20E18",
      "20F45"
    ],
    "keywords": [
      "compact groups",
      "engel profinite groups",
      "right engel element",
      "locally nilpotent",
      "finite normal subgroup"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}