{
  "id": "1110.6888",
  "version": "v1",
  "published": "2011-10-31T18:31:15.000Z",
  "updated": "2011-10-31T18:31:15.000Z",
  "title": "Finite $p$-groups of class 3 with noninner automorphisms of order $p$",
  "authors": [
    "Alireza Abdollahi",
    "Mohsen Ghoraishi"
  ],
  "categories": [
    "math.GR"
  ],
  "abstract": "A longstanding conjecture asserts that every non-abelian finite $p$-group $G$ admits a non-inner automorphism of order $p$. The conjecture is valid for finite $p$-groups of class 2. Here, we prove every finite non-abelian $p$-group $G$ of class 3 with $p>2$ has a noninner automorphism of order $p$ leaving $\\Phi(G)$ elementwise fixed. We also prove that if $G$ is a finite 2-group of class 3 which cannot be generated by 4 elements, then $G$ has a non-inner automorphism of order 2 leaving $\\Phi(G)$ elementwise fixed. We also prove that the latter conclusion holds for finite 2-groups $G$ of class 3 such that the center of $G$ is not cyclic and the minimal number of generators of $G$ is 2 or 4 and it holds whenever the center of $G$ is {\\em not} 2-generated and the minimal number of generators of $G$ is 3. Some results are also proved for the existence of non-inner automorphisms of order $p$ for a finite $p$-group $G$ under conditions in terms of the minimal number of generators of the center factor of $G$ and a certain function of the rank of $G$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-10-31T18:31:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20D15",
      "20D45"
    ],
    "keywords": [
      "noninner automorphism",
      "non-inner automorphism",
      "minimal number",
      "generators",
      "longstanding conjecture asserts"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1110.6888A"
    }
  }
}