{
  "id": "2008.10800",
  "version": "v1",
  "published": "2020-08-25T03:39:47.000Z",
  "updated": "2020-08-25T03:39:47.000Z",
  "title": "Virtually nilpotent groups with finitely many orbits under automorphisms",
  "authors": [
    "Raimundo Bastos",
    "Alex C. Dantas",
    "Emerson de Melo"
  ],
  "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 $\\omega(G)$. Let $G$ be a virtually nilpotent group such that $\\omega(G)< \\infty$. We prove that $G = K \\rtimes H$ where $H$ is a torsion subgroup and $K$ is a torsion-free nilpotent radicable characteristic subgroup of $G$. Moreover, we prove that $G^{'}= D \\times \\Tor(G^{'})$ where $D$ is a torsion-free nilpotent radicable characteristic subgroup. In particular, if the maximum normal torsion subgroup $\\tau(G)$ of $G$ is trivial, then $G^{'}$ is nilpotent.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-08-25T03:39:47.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20E22",
      "20E36"
    ],
    "keywords": [
      "virtually nilpotent group",
      "torsion-free nilpotent radicable characteristic subgroup",
      "automorphism orbits",
      "maximum normal torsion subgroup",
      "natural action"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}