{
  "id": "1908.01375",
  "version": "v1",
  "published": "2019-08-04T17:14:43.000Z",
  "updated": "2019-08-04T17:14:43.000Z",
  "title": "Soluble Groups with few orbits under automorphisms",
  "authors": [
    "Raimundo Bastos",
    "Alex Carrazedo Dantas",
    "Emerson de Melo"
  ],
  "comment": "Submitted to an international 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 $\\omega(G)$. We prove that if $G$ is a soluble group with finite rank such that $\\omega(G)< \\infty$, then $G$ contains a torsion-free characteristic nilpotent subgroup $K$ such that $G = K \\rtimes H$, where $H$ is a finite group. Moreover, we classify the mixed order soluble groups of finite rank such that $\\omega(G)=3$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-08-04T17:14:43.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20E22",
      "20E36"
    ],
    "keywords": [
      "automorphism orbits",
      "finite rank",
      "torsion-free characteristic nilpotent subgroup",
      "mixed order soluble groups",
      "natural action"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}