{
  "id": "1806.11132",
  "version": "v1",
  "published": "2018-06-28T18:09:06.000Z",
  "updated": "2018-06-28T18:09:06.000Z",
  "title": "FC-groups with finitely many automorphism orbits",
  "authors": [
    "Raimundo A. Bastos",
    "Alex C. Dantas"
  ],
  "comment": "Submitted to an internacional journal",
  "categories": [
    "math.GR"
  ],
  "abstract": "Let $G$ be a group. The orbits of the natural action of $Aut(G)$ on $G$ are called \"automorphism orbits\" of $G$, and the number of automorphism orbits of $G$ is denoted by $\\omega(G)$. In this paper we prove that if $G$ is an FC-group with finitely many automorphism orbits, then the derived subgroup $G'$ is finite and $G$ admits a decomposition $G = Tor(G) \\times D$, where $Tor(G)$ is the torsion subgroup of $G$ and $D$ is a divisible characteristic subgroup of $Z(G)$. We also show that if $G$ is an infinite FC-group with $\\omega(G) \\leqslant 8$, then either $G$ is soluble or $G \\cong A_5 \\times H$, where $H$ is an infinite abelian group with $\\omega(H)=2$. Moreover, we describe the structure of the infinite non-soluble FC-groups with at most eleven automorphism orbits.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-06-28T18:09:06.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20E36",
      "20F24"
    ],
    "keywords": [
      "automorphism orbits",
      "infinite abelian group",
      "torsion subgroup",
      "infinite non-soluble fc-groups",
      "infinite fc-group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}