{
  "id": "1512.01459",
  "version": "v1",
  "published": "2015-12-04T15:47:05.000Z",
  "updated": "2015-12-04T15:47:05.000Z",
  "title": "On the lattice of subracks of the rack of a finite group",
  "authors": [
    "Istvan Heckenberger",
    "John Shareshian",
    "Volkmar Welker"
  ],
  "categories": [
    "math.CO",
    "math.GR",
    "math.QA"
  ],
  "abstract": "In this paper we initiate the study of racks from the combined perspective of combinatorics and finite group theory. A rack R is a set with a self-distributive binary operation. We study the combinatorics of the partially ordered set {\\cal R}(R) of all subracks of R with inclusion as the order relation. Groups G with the conjugation operation provide an important class of racks. For the case R = G we show that -> the order complex of {\\cal R}(R) has the homotopy type of a sphere, -> the isomorphism type of {\\cal R}(R) determines if G is abelian, nilpotent, supersolvable, solvable or simple, -> {\\cal R}(R) is graded if and only if G is abelian, G = S_3, G = D_8 or G = Q_8. In addition, we provide some examples of subracks R of a group G for which {\\cal R}(R) relates to well studied combinatorial structures. In particular, the examples show that the order complex of {\\cal R}(R) for general R is more complicated than in the case R = G.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-12-04T15:47:05.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E45",
      "20D30"
    ],
    "keywords": [
      "order complex",
      "finite group theory",
      "order relation",
      "combinatorics",
      "conjugation operation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv151201459H"
    }
  }
}