{
  "id": "2302.03947",
  "version": "v1",
  "published": "2023-02-08T08:51:08.000Z",
  "updated": "2023-02-08T08:51:08.000Z",
  "title": "Diameter of a direct power of alternating groups",
  "authors": [
    "A. Azad",
    "N. Karimi"
  ],
  "comment": "arXiv admin note: text overlap with arXiv:1506.02695",
  "categories": [
    "math.GR"
  ],
  "abstract": "So far, it has been proven that if $G$ is an abelian group , then the diameter of $G^n$ with respect to any generating set is $O(n)$; and if $G$ is nilpotent, symmetric or dihedral, then there exists a generating set of minimum size, for which the diameter of $G^n$ is $O(n)$ \\cite{Karimi:2017}. In \\cite{Dona:2022} it has been proven that if $G$ is a non-abelian simple group, then the diameter of $G^n$ with respect to any generating set is $O(n^3)$. In this paper we estimate the diameter of direct power of alternating groups $A_n$ for $n \\geq 4$, i.e. a class of non-abelian simple groups. We show that there exist a generating set of minimum size for $A_4^n$, for which the diameter of $A_4^n$ is $O(n)$. For $n \\geq 5$, we show that there exists a generating set of minimum size for $A_n^2$, for which the diameter of $A_n^2$ is at most $O(ne^{(c+1) (\\log \\,n)^4 \\log \\log n})$ , for an absolute constant $c >0$. Finally for $ 1\\leq n \\leq 8 $, we provide generating sets of size two for $A_5^n$ and we show that the diameter of $A_5^n$ with respect to those generating sets is $O(n)$. These results are more pieces of evidence for a conjecture which has been presented in \\cite{Karimithesis:2015} in 2015.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-02-08T08:51:08.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "generating set",
      "direct power",
      "alternating groups",
      "non-abelian simple group",
      "minimum size"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}