{
  "id": "1205.0450",
  "version": "v3",
  "published": "2012-05-02T15:02:36.000Z",
  "updated": "2012-10-04T09:49:48.000Z",
  "title": "The classification of normalizing groups",
  "authors": [
    "João Araújo",
    "Peter J. Cameron",
    "James Mitchell",
    "Max Neunhöffer"
  ],
  "comment": "Three lemmas replaced with a shorter argument and other changes suggested by the referee of the Journal of Algebra",
  "categories": [
    "math.GR"
  ],
  "abstract": "Let $X$ be a finite set such that $|X|=n$. Let $\\trans$ and $\\sym$ denote respectively the transformation monoid and the symmetric group on $n$ points. Given $a\\in \\trans\\setminus \\sym$, we say that a group $G\\leq \\sym$ is $a$-normalizing if $$<a,G> \\setminus G=<g^{{-1}}ag\\mid g\\in G>.$$ If $G$ is $a$-normalizing for all $a\\in \\trans\\setminus \\sym$, then we say that $G$ is normalizing. The goal of this paper is to classify normalizing groups and hence answer a question posed elsewhere. The paper ends with a number of problems for experts in groups, semigroups and matrix theory.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2012-10-04T09:49:48.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "classification",
      "finite set",
      "transformation monoid",
      "symmetric group",
      "paper ends"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1205.0450A"
    }
  }
}