{
  "id": "1908.00331",
  "version": "v1",
  "published": "2019-08-01T11:20:35.000Z",
  "updated": "2019-08-01T11:20:35.000Z",
  "title": "Orbits in Extra-special $p$-Groups for $p$ an Odd Prime",
  "authors": [
    "C P Anil Kumar",
    "Soham Swadhin Pradhan"
  ],
  "comment": "21 pages",
  "categories": [
    "math.GR"
  ],
  "abstract": "For an odd prime $p$ and a positive integer $n$, it is well known that there are two types of extra-special $p$-groups of order $p^{2n+1}$, first one is the Heisenberg group which has exponent $p$ and the second one is of exponent $p^2$. In this article, a new way of representing the extra-special $p$-group of exponent $p^2$ is given, which is suggested in a natural way by a familiar representation of the Heisenberg group. These representations facilitate an explicit way of finding formulae for any automorphism and any endomorphism of an extra-special $p$-group $G$ for both the types. Based on these formulae, the automorphism group $Aut(G)$ and the endomorphism semigroup $End(G)$ are described. The orbits under the action of the automorphism group $Aut(G)$ are determined. In addition, the endomorphism semigroup image of any element in $G$ is found. As a consequence it is deduced that, under the notion of degeneration of elements in $G$, the endomorphism semigroup $End(G)$ induces a partial order on the automorphism orbits when $G$ is the Heisenberg group and does not induce when $G$ is the extra-special $p$-group of exponent $p^2$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-08-01T11:20:35.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20D15"
    ],
    "keywords": [
      "odd prime",
      "extra-special",
      "heisenberg group",
      "automorphism group",
      "endomorphism semigroup image"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}