{
  "id": "1505.05622",
  "version": "v1",
  "published": "2015-05-21T07:17:04.000Z",
  "updated": "2015-05-21T07:17:04.000Z",
  "title": "On Equality of Certain Automorphism Groups",
  "authors": [
    "Surjeet Kour",
    "Vishakha"
  ],
  "categories": [
    "math.GR"
  ],
  "abstract": "Let $G = H\\times A$ be a finite group, where $H$ is a purely non-abelian subgroup of $G$ and $A$ is a non-trivial abelian factor of $G$. Then, for $n \\geq 2$, we show that there exists an isomorphism $\\phi : Aut_{Z(G)}^{\\gamma_{n}(G)}(G) \\rightarrow Aut_{Z(H)}^{\\gamma_{n}(H)}(H)$ such that $\\phi(Aut_{c}^{n-1}(G))=Aut_{c}^{n-1}(H)$. We also give some necessary and sufficient conditions on a finite $p$-group $G$ such that $Autcent(G)=Aut_{c}^{n-1}(G)$ . Furthermore, for a finite non-abelian $p$-group $G$, we give some necessary and sufficient conditions for $Aut_{Z(G)}^{\\gamma_{n}(G)}(G)$ to be equal to $Aut_{\\gamma_2(G)}^{Z(G)}(G)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-05-21T07:17:04.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20D15",
      "20D45"
    ],
    "keywords": [
      "automorphism groups",
      "sufficient conditions",
      "non-trivial abelian factor",
      "purely non-abelian subgroup",
      "finite group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}