{
  "id": "1010.0343",
  "version": "v1",
  "published": "2010-10-02T14:57:41.000Z",
  "updated": "2010-10-02T14:57:41.000Z",
  "title": "Frobenius groups of automorphisms and their fixed points",
  "authors": [
    "Evgenii I. Khukhro",
    "Natalia Yu. Makarenko",
    "Pavel Shumyatsky"
  ],
  "comment": "31 pages",
  "doi": "10.1515/form.2011.152",
  "categories": [
    "math.GR",
    "math.RA"
  ],
  "abstract": "Suppose that a finite group $G$ admits a Frobenius group of automorphisms $FH$ with kernel $F$ and complement $H$ such that the fixed-point subgroup of $F$ is trivial: $C_G(F)=1$. In this situation various properties of $G$ are shown to be close to the corresponding properties of $C_G(H)$. By using Clifford's theorem it is proved that the order $|G|$ is bounded in terms of $|H|$ and $|C_G(H)|$, the rank of $G$ is bounded in terms of $|H|$ and the rank of $C_G(H)$, and that $G$ is nilpotent if $C_G(H)$ is nilpotent. Lie ring methods are used for bounding the exponent and the nilpotency class of $G$ in the case of metacyclic $FH$. The exponent of $G$ is bounded in terms of $|FH|$ and the exponent of $C_G(H)$ by using Lazard's Lie algebra associated with the Jennings--Zassenhaus filtration and its connection with powerful subgroups. The nilpotency class of $G$ is bounded in terms of $|H|$ and the nilpotency class of $C_G(H)$ by considering Lie rings with a finite cyclic grading satisfying a certain `selective nilpotency' condition. The latter technique also yields similar results bounding the nilpotency class of Lie rings and algebras with a metacyclic Frobenius group of automorphisms, with corollaries for connected Lie groups and torsion-free locally nilpotent groups with such groups of automorphisms. Examples show that such nilpotency results are no longer true for non-metacyclic Frobenius groups of automorphisms.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-10-02T14:57:41.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "17B40",
      "20D45",
      "17B70",
      "20D15",
      "20E36",
      "20F40",
      "22E25"
    ],
    "keywords": [
      "automorphisms",
      "nilpotency class",
      "fixed points",
      "non-metacyclic frobenius groups",
      "torsion-free locally nilpotent groups"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 31,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1010.0343K"
    }
  }
}