{
  "id": "1707.04643",
  "version": "v1",
  "published": "2017-07-14T21:19:21.000Z",
  "updated": "2017-07-14T21:19:21.000Z",
  "title": "Set-Direct Factorizations of Groups",
  "authors": [
    "Dan Levy",
    "Attila Maróti"
  ],
  "comment": "19 pages",
  "categories": [
    "math.GR"
  ],
  "abstract": "We consider factorizations $G=XY$ where $G$ is a general group, $X$ and $Y$ are normal subsets of $G$ and any $g\\in G$ has a unique representation $g=xy$ with $x\\in X$ and $y\\in Y$. This definition coincides with the customary and extensively studied definition of a direct product decomposition by subsets of a finite abelian group. Our main result states that a group $G$ has such a factorization if and only if $G$ is a central product of $\\left\\langle X\\right\\rangle $ and $\\left\\langle Y\\right\\rangle $ and the central subgroup $\\left\\langle X\\right\\rangle \\cap \\left\\langle Y\\right\\rangle $ satisfies certain abelian factorization conditions. We analyze some special cases and give examples. In particular, simple groups have no non-trivial set-direct factorization.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-07-14T21:19:21.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20K25",
      "20D40"
    ],
    "keywords": [
      "main result states",
      "non-trivial set-direct factorization",
      "abelian factorization conditions",
      "finite abelian group",
      "direct product decomposition"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}