{
  "id": "2209.04948",
  "version": "v1",
  "published": "2022-09-11T22:12:41.000Z",
  "updated": "2022-09-11T22:12:41.000Z",
  "title": "Construction of All Gyrogroups of Orders at most 31",
  "authors": [
    "Ali Reza Ashrafi",
    "Kurosh Mavaddat Nezhaad",
    "Mohammad Ali Salahshour"
  ],
  "comment": "9 pages",
  "categories": [
    "math.GR"
  ],
  "abstract": "The gyrogroup is the closest algebraic structure to the group ever discovered. It has a binary operation $\\star$ containing an identity element such that each element has an inverse. Furthermore, for each pair $(a,b)$ of elements of this structure there exists an automorphism $\\gyr{a,b}{}$ with this property that left associativity and left loop property are satisfied. Since each gyrogroup is a left Bol loop, some results of Burn imply that all gyrogroups of orders $p, 2p$ and $p^2$ are groups. The aim of this paper is to classify gyrogroups of orders 8, 12, 15, 18, 20, 21, and 28.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-09-11T22:12:41.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20N05"
    ],
    "keywords": [
      "construction",
      "closest algebraic structure",
      "left bol loop",
      "left loop property",
      "identity element"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}