{
  "id": "2411.03846",
  "version": "v2",
  "published": "2024-11-06T11:33:10.000Z",
  "updated": "2024-12-20T15:57:06.000Z",
  "title": "Transfinite hypercentral iterated wreath product of integral domains",
  "authors": [
    "Riccardo Aragona",
    "Norberto Gavioli",
    "Giuseppe Nozzi"
  ],
  "comment": "This version is a major revision of the previous one, with corrections and new results",
  "categories": [
    "math.GR",
    "math.CO"
  ],
  "abstract": "Starting with an integral domain $D$ of characteristic $0$, we consider a class of iterated wreath product $W_n$ of $n$ copies of $D$. In order that $W_n$ be transfinite hypercentral, it is necessary to restrict to the case of wreath products defined by way of numerical polynomials. We also associate to each of these groups a Lie ring, providing a correspondence preserving most of the structure. This construction generalizes a result which characterizes the Lie algebras associated to the Sylow $p$-subgroups of the symmetric group $\\mathrm{Sym}(p^n)$. As an application, we explore the normalizer chain $\\lbrace\\mathbf{N}_{i}\\rbrace_{i\\geq -1}$ starting from the canonical regular abelian subgroup $T$ of $W_n$. Finally, we characterize the regular abelian normal subgroups of $\\mathbf{N}_0$ that are isomorphic to $D^n$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2024-12-20T15:57:06.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20E22",
      "20B35",
      "20E15",
      "20F19",
      "05A17"
    ],
    "keywords": [
      "transfinite hypercentral iterated wreath product",
      "integral domain",
      "regular abelian normal subgroups",
      "canonical regular abelian subgroup",
      "construction generalizes"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}