{
  "id": "1512.06097",
  "version": "v1",
  "published": "2015-12-18T20:21:14.000Z",
  "updated": "2015-12-18T20:21:14.000Z",
  "title": "Almost Engel finite and profinite groups",
  "authors": [
    "E. I. Khukhro",
    "P. Shumyatsky"
  ],
  "categories": [
    "math.GR"
  ],
  "abstract": "Let $g$ be an element of a group $G$. For a positive integer $n$, let $E_n(g)$ be the subgroup generated by all commutators $[...[[x,g],g],\\dots ,g]$ over $x\\in G$, where $g$ is repeated $n$ times. We prove that if $G$ is a profinite group such that for every $g\\in G$ there is $n=n(g)$ such that $E_n(g)$ is finite, then $G$ has a finite normal subgroup $N$ such that $G/N$ is locally nilpotent. The proof uses the Wilson--Zelmanov theorem saying that Engel profinite groups are locally nilpotent. In the case of a finite group $G$, we prove that if, for some $n$, $|E_n(g)|\\leq m$ for all $g\\in G$, then the order of the nilpotent residual $\\gamma _{\\infty}(G)$ is bounded in terms of $m$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-12-18T20:21:14.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "engel finite",
      "engel profinite groups",
      "finite normal subgroup",
      "locally nilpotent",
      "nilpotent residual"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv151206097K"
    }
  }
}