{
  "id": "0808.0308",
  "version": "v3",
  "published": "2008-08-03T10:25:18.000Z",
  "updated": "2011-01-25T06:41:32.000Z",
  "title": "Notes on periodic elements of Garside groups",
  "authors": [
    "Eon-Kyung Lee",
    "Sang-Jin Lee"
  ],
  "comment": "The contents of this 8-page paper have been subsumed into the 27-page paper, \"Periodic elements in Garside groups\" (arXiv:1004.5308)",
  "categories": [
    "math.GT",
    "math.GR"
  ],
  "abstract": "Let $G$ be a Garside group with Garside element $\\Delta$. An element $g$ in $G$ is said to be \\emph{periodic} if some power of $g$ lies in the cyclic group generated by $\\Delta$. This paper shows the following. (i) The periodicity of an element does not depend on the choice of a particular Garside structure if and only if the center of $G$ is cyclic. (ii) If $g^k=\\Delta^{ka}$ for some nonzero integer $k$, then $g$ is conjugate to $\\Delta^a$. (iii) Every finite subgroup of the quotient group $G/<\\Delta^m>$ is cyclic, where $\\Delta^m$ is the minimal positive central power of $\\Delta$.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2011-01-25T06:41:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F36",
      "20F10"
    ],
    "keywords": [
      "garside group",
      "periodic elements",
      "minimal positive central power",
      "garside element",
      "cyclic group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0808.0308L"
    }
  }
}