{
  "id": "1712.06516",
  "version": "v1",
  "published": "2017-12-18T16:51:28.000Z",
  "updated": "2017-12-18T16:51:28.000Z",
  "title": "Automorphisms of dihedral-like automorphic loops",
  "authors": [
    "Mouna Aboras",
    "Petr Vojtěchovský"
  ],
  "journal": "Communications in Algebra 44 (2016), no. 2, 613-627",
  "categories": [
    "math.GR"
  ],
  "abstract": "Automorphic loops are loops in which all inner mappings are automorphisms. A large class of automorphic loops is obtained as follows: Let $m$ be a positive even integer, $G$ an abelian group, and $\\alpha$ an automorphism of $G$ that satisfies $\\alpha^2=1$ if $m>2$. Then the dihedral-like automorphic loop $\\mathrm{Dih}(m,G,\\alpha)$ is defined on $\\mathbb Z_m\\times G$ by $(i,u)(j,v)=(i+j, ((-1)^{j}u+v)\\alpha^{ij})$. We prove that two finite dihedral-like automorphic loops $\\mathrm{Dih}(m,G,\\alpha)$, $\\mathrm{Dih}(\\overline{m},\\overline{G},\\overline{\\alpha})$ are isomorphic if and only if $m=\\overline{m}$, $G=\\overline{G}$, and $\\alpha$ is conjugate to $\\overline{\\alpha}$ in the automorphism group of $G$. Moreover, for a finite dihedral-like automorphic loop $Q$ we describe the structure of the automorphism group of $Q$ and its subgroup consisting of inner mappings of $Q$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-12-18T16:51:28.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20N05",
      "20D45"
    ],
    "keywords": [
      "finite dihedral-like automorphic loop",
      "inner mappings",
      "automorphism group",
      "abelian group",
      "large class"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}