{
  "id": "1902.02154",
  "version": "v1",
  "published": "2019-02-06T13:10:49.000Z",
  "updated": "2019-02-06T13:10:49.000Z",
  "title": "On embeddings of quandles into groups",
  "authors": [
    "Valeriy Bardakov",
    "Timur Nasybullov"
  ],
  "comment": "19 pages",
  "categories": [
    "math.GR",
    "math.GT"
  ],
  "abstract": "In the present paper, we introduce the new construction of quandles. For a group $G$ and its subset $A$ we construct a quandle $Q(G,A)$ which is called the $(G,A)$-quandle and study properties of this quandle. In particular, we prove that if $Q$ is a quandle such that the natural map $Q\\to G_Q$ from $Q$ to its enveloping group $G_Q$ is injective, then $Q$ is the $(G,A)$-quandle for an appropriate group $G$ and its subset $A$. Also we introduce the free product of quandles and study this construction for $(G,A)$-quandles. In addition, we classify all finite quandles with enveloping group $\\mathbb{Z}^2$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-02-06T13:10:49.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20N02",
      "57M27"
    ],
    "keywords": [
      "embeddings",
      "enveloping group",
      "free product",
      "construction",
      "study properties"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}