{
  "id": "math/0509505",
  "version": "v2",
  "published": "2005-09-22T03:09:22.000Z",
  "updated": "2005-10-08T13:35:04.000Z",
  "title": "The Number of Finite Groups Whose Element Orders is Given",
  "authors": [
    "A. R. Moghaddamfar",
    "W. J. Shi"
  ],
  "comment": "17 pages",
  "categories": [
    "math.GR"
  ],
  "abstract": "The spectrum $\\omega(G)$ of a finite group $G$ is the set of element orders of $G$. If $\\Omega$ is a non-empty subset of the set of natural numbers, $h(\\Omega)$ stands for the number of isomorphism classes of finite groups $G$ with $\\omega(G)=\\Omega$ and put $h(G)=h(\\omega(G))$. We say that $G$ is recognizable (by spectrum $\\omega(G)$) if $h(G)=1$. The group $G$ is almost recognizable (resp. nonrecognizable) if $1<h(G)<\\infty$ (resp. $h(G)=\\infty$). In the present paper, we focus our attention on the projective general linear groups ${PGL}(2,p^n)$, where $p=2^\\alpha 3^\\beta+1$ is a prime, $\\alpha \\geq 0, \\beta \\geq 0$ and $n\\geq 1$, and we show that these groups cannot be almost recognizable, in other words $h({PGL}(2,p^n))\\in \\{1, \\infty\\}$. It is also shown that the projective general linear groups ${PGL}(2,7)$ and ${PGL}(2,9)$ are nonrecognizable. In this paper a computer program has also been presented in order to find out the primitive prime divisors of $a^n-1$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2005-10-08T13:35:04.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20D05"
    ],
    "keywords": [
      "finite group",
      "element orders",
      "projective general linear groups",
      "isomorphism classes",
      "natural numbers"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math......9505M"
    }
  }
}