{
  "id": "math/0602039",
  "version": "v5",
  "published": "2006-02-02T16:58:21.000Z",
  "updated": "2006-10-09T18:55:06.000Z",
  "title": "The Automorphism Group of a Finite p-Group is Almost Always a p-Group",
  "authors": [
    "Geir T. Helleloid",
    "Ursula Martin"
  ],
  "comment": "38 pages, to appear in the Journal of Algebra; improved references, changes in terminology",
  "journal": "Journal of Algebra 312(2007) 294-329",
  "doi": "10.1016/j.jalgebra.2007.01.008",
  "categories": [
    "math.GR",
    "math.CO"
  ],
  "abstract": "Many common finite p-groups admit automorphisms of order coprime to p, and when p is odd, it is reasonably difficult to find finite p-groups whose automorphism group is a p-group. Yet the goal of this paper is to prove that the automorphism group of a finite p-group is almost always a p-group. The asymptotics in our theorem involve fixing any two of the following parameters and letting the third go to infinity: the lower p-length, the number of generators, and p. The proof of this theorem depends on a variety of topics: counting subgroups of a p-group; analyzing the lower p-series of a free group via its connection with the free Lie algebra; counting submodules of a module via Hall polynomials; and using numerical estimates on Gaussian coefficients.",
  "revisions": [
    {
      "version": "v5",
      "updated": "2006-10-09T18:55:06.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20E36",
      "05A16"
    ],
    "keywords": [
      "automorphism group",
      "common finite p-groups admit automorphisms",
      "free lie algebra",
      "lower p-series",
      "lower p-length"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 38,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math......2039H"
    }
  }
}