{
  "id": "math/0502512",
  "version": "v5",
  "published": "2005-02-24T12:40:43.000Z",
  "updated": "2006-10-06T18:46:01.000Z",
  "title": "On infinite groups generated by two quaternions",
  "authors": [
    "Diego Rattaggi"
  ],
  "comment": "31 pages. Completely revised version, several new results and simplified proofs",
  "categories": [
    "math.GR",
    "math.RA"
  ],
  "abstract": "Let $x$, $y$ be two integral quaternions of norm $p$ and $l$, respectively, where $p$, $l$ are distinct odd prime numbers. We investigate the structure of $<x,y>$, the multiplicative group generated by $x$ and $y$. Under a certain condition which excludes $<x,y>$ from being free or abelian, we show for example that $<x,y>$, its center, commutator subgroup and abelianization are finitely presented infinite groups. We give many examples where our condition is satisfied and compute as an illustration a finite presentation of the group $<1+j+k, 1+2j>$ having these two generators and seven relations. In a second part, we study the basic question whether there exist commuting quaternions $x$ and $y$ for fixed $p$, $l$, using results on prime numbers of the form $r^2 + m s^2$ and a simple invariant for commutativity.",
  "revisions": [
    {
      "version": "v5",
      "updated": "2006-10-06T18:46:01.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "infinite groups",
      "distinct odd prime numbers",
      "integral quaternions",
      "simple invariant",
      "commutator subgroup"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 31,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math......2512R"
    }
  }
}