{
  "id": "1705.10607",
  "version": "v1",
  "published": "2017-05-30T13:18:56.000Z",
  "updated": "2017-05-30T13:18:56.000Z",
  "title": "Automorphism groups of quandles and related groups",
  "authors": [
    "Valeriy Bardakov",
    "Timur Nasybullov",
    "Mahender Singh"
  ],
  "comment": "19 pages",
  "categories": [
    "math.GR",
    "math.GT"
  ],
  "abstract": "In this paper we study different questions concerning automorphisms of quandles. For a conjugation quandle $Q={\\rm Conj}(G)$ of a group $G$ we determine several subgroups of ${\\rm Aut}(Q)$ and find necessary and sufficient conditions when these subgroups coincide with the whole group ${\\rm Aut}(Q)$. In particular, we prove that ${\\rm Aut}({\\rm Conj}(G))={\\rm Z}(G)\\rtimes {\\rm Aut}(G)$ if and only if either ${\\rm Z}(G)=1$ or $G$ is one of the groups $\\mathbb{Z}_2$, $\\mathbb{Z}_2^2$ or $\\mathbb{Z}_3$. For a big list of Takasaki quandles $T(G)$ of an abelian group $G$ with $2$-torsion we prove that the group of inner automorphisms ${\\rm Inn}(T(G))$ is a Coxeter group. We study automorphisms of certain extensions of quandles and determine some interesting subgroups of the automorphism groups of these quandles. Also we classify finite quandles $Q$ with $3\\leq k$-transitive action of ${\\rm Aut}(Q)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-05-30T13:18:56.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20N02",
      "20B25",
      "57M27"
    ],
    "keywords": [
      "automorphism groups",
      "related groups",
      "questions concerning automorphisms",
      "sufficient conditions",
      "conjugation quandle"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}