{
  "id": "math/0701711",
  "version": "v1",
  "published": "2007-01-24T19:01:44.000Z",
  "updated": "2007-01-24T19:01:44.000Z",
  "title": "C-loops: An introduction",
  "authors": [
    "J. D. Phillips",
    "Petr Vojtěchovský"
  ],
  "comment": "15 pages",
  "journal": "Publicationes Mathematicae Debrecen 68 (2006), nos. 1-2, 115-137",
  "categories": [
    "math.GR"
  ],
  "abstract": "C-loops are loops satisfying $x(y(yz))=((xy)y)z$. They often behave analogously to Moufang loops and they are closely related to Steiner triple systems and combinatorics. We initiate the study of C-loops by proving: (i) Steiner loops are C-loops, (ii) C-loops are alternative, inverse property loops with squares in the nucleus, (iii) the nucleus of a C-loop is a normal subgroup, (iv) C-loops modulo their nucleus are Steiner loops, (v) C-loops are power associative, power alternative but not necessarily diassociative, (vi) torsion commutative C-loops are products of torsion abelian groups and torsion commutative 2-C-loops; and several other results. We also give examples of the smallest nonassociative C-loops, and explore the analogy between commutative C-loops and commutative Moufang loops.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2007-01-24T19:01:44.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20N05"
    ],
    "keywords": [
      "introduction",
      "steiner loops",
      "torsion abelian groups",
      "steiner triple systems",
      "inverse property loops"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007math......1711P"
    }
  }
}