{
  "id": "1707.06889",
  "version": "v1",
  "published": "2017-07-20T10:01:20.000Z",
  "updated": "2017-07-20T10:01:20.000Z",
  "title": "Profinite groups and centralizers of coprime automorphisms whose elements are Engel",
  "authors": [
    "Cristina Acciarri",
    "Danilo Sanção da Silveira"
  ],
  "comment": "arXiv admin note: text overlap with arXiv:1702.02899, arXiv:1602.01661, arXiv:1108.0698",
  "categories": [
    "math.GR"
  ],
  "abstract": "Let $q$ be a prime, $n$ a positive integer and $A$ an elementary abelian group of order $q^r$ with $r\\geq2$ acting on a finite $q'$-group $G$. The following results are proved. We show that if all elements in $\\gamma_{r-1}(C_G(a))$ are $n$-Engel in $G$ for any $a\\in A^\\#$, then $\\gamma_{r-1}(G)$ is $k$-Engel for some $\\{n,q,r\\}$-bounded number $k$, and if, for some integer $d$ such that $2^d\\leq r-1$, all elements in the $d$th derived group of $C_G(a)$ are $n$-Engel in $G$ for any $a\\in A^\\#$, then the $d$th derived group $G^{(d)}$ is $k$-Engel for some $\\{n,q,r\\}$-bounded number $k$. Assuming $r\\geq 3$ we prove that if all elements in $\\gamma_{r-2}(C_G(a))$ are $n$-Engel in $C_G(a)$ for any $a\\in A^\\#$, then $\\gamma_{r-2}(G)$ is $k$-Engel for some $\\{n,q,r\\}$-bounded number $k$, and if, for some integer $d$ such that $2^d\\leq r-2$, all elements in the $d$th derived group of $C_G(a)$ are $n$-Engel in $C_G(a)$ for any $a\\in A^\\#,$ then the $d$th derived group $G^{(d)}$ is $k$-Engel for some $\\{n,q,r\\}$-bounded number $k$. Analogue (non-quantitative) results for profinite groups are also obtained.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-07-20T10:01:20.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20D45",
      "20E18",
      "20F40",
      "20F45"
    ],
    "keywords": [
      "th derived group",
      "profinite groups",
      "coprime automorphisms",
      "bounded number",
      "centralizers"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}