{
  "id": "math/0302120",
  "version": "v2",
  "published": "2003-02-11T14:33:07.000Z",
  "updated": "2004-01-14T02:16:52.000Z",
  "title": "On the holomorph of a discrete group",
  "authors": [
    "Maria S. Voloshina"
  ],
  "comment": "87 pages, 12 figures. Ph.D. dissertation, University of Rochester. Advisor: Frederick R. Cohen",
  "categories": [
    "math.GR",
    "math.AT"
  ],
  "abstract": "The holomorph of a discrete group $G$ is the universal semi-direct product of $G$. In chapter 1 we describe why it is an interesting object and state main results. In chapter 2 we recall the classical definition of the holomorph as well as this universal property, and give some group theoretic properties and examples of holomorphs. In particular, we give a necessary and sufficient condition for the existence of a map of split extensions for holomorphs of two groups. In chapter 3 we construct a resolution for $Hol(Z_{p^r})$ for every prime $p$, where ${\\mathbb Z}_m$ denotes a cyclic group of order $m$, and use it to compute the integer homology and mod $p$ cohomology ring of $Hol(Z_{p^r})$. In chapter 4 we study the holomorph of the direct sum of several copies of $Z_{p^r}$. We identify this holomorph as a nice subgroup of $GL(n+1, Z_{p^r})$, thus its cohomology informs on the cohomology of the general linear group which has been of interest in the subject. We show that the LHS spectral sequence for $H^*(Hol(\\bigoplus_n Z_{p^r}); F_p)$ does not collapse at the $E_2$ stage for $p^r\\ge 8$. Also, we compute mod $p$ cohomology and the first Bockstein homomorphisms of the congruence subgroups given by $Ker (Hol(\\bigoplus_n Z_{p^r}) \\to Hol(\\bigoplus_n Z_p)).$ In chapter 5 we recall wreath products and permutative categories, and their connections with holomorphs. In chapter 6 we give a short proof of the well-known fact due to S. Eilenberg and J. C. Moore that the only injective object in the category of groups is the trivial group.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2004-01-14T02:16:52.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20J06",
      "20E22",
      "20J15",
      "20G40",
      "20G10",
      "18G05",
      "18G40"
    ],
    "keywords": [
      "discrete group",
      "cohomology",
      "state main results",
      "group theoretic properties",
      "first bockstein homomorphisms"
    ],
    "tags": [
      "dissertation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 87,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2003math......2120V"
    }
  }
}