{
  "id": "1809.07191",
  "version": "v1",
  "published": "2018-09-19T13:43:53.000Z",
  "updated": "2018-09-19T13:43:53.000Z",
  "title": "Characterizations of Cancellable Groups",
  "authors": [
    "Matthew Harrison-Trainor",
    "Meng-Che \"Turbo\" Ho"
  ],
  "comment": "14 pages",
  "categories": [
    "math.LO",
    "math.GR"
  ],
  "abstract": "An abelian group $A$ is said to be cancellable if whenever $A \\oplus G$ is isomorphic to $A \\oplus H$, $G$ is isomorphic to $H$. We show that the index set of cancellable rank 1 torsion-free abelian groups is $\\Pi^0_4$ $m$-complete, showing that the classification by Fuchs and Loonstra cannot be simplified. For arbitrary non-finitely generated groups, we show that the cancellation property is $\\Pi^1_1$ $m$-hard; we know of no upper bound, but we conjecture that it is $\\Pi^1_2$ $m$-complete.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-09-19T13:43:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03D80",
      "20Kxx",
      "20K25"
    ],
    "keywords": [
      "cancellable groups",
      "characterizations",
      "torsion-free abelian groups",
      "isomorphic",
      "upper bound"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}