{
  "id": "2505.03017",
  "version": "v1",
  "published": "2025-05-05T20:37:51.000Z",
  "updated": "2025-05-05T20:37:51.000Z",
  "title": "Local--global generation property of commutators in finite $π$-soluble groups",
  "authors": [
    "Cristina Acciarri",
    "Robert M. Guralnick",
    "Evgeny Khukhro",
    "Pavel Shumyatsky"
  ],
  "comment": "arXiv admin note: text overlap with arXiv:2404.14599",
  "categories": [
    "math.GR"
  ],
  "abstract": "For a group $A$ acting by automorphisms on a group $G$, let $I_G(A)$ denote the set of commutators $[g,a]=g^{-1}g^a$, where $g\\in G$ and $a\\in A$, so that $[G,A]$ is the subgroup generated by $I_G(A)$. We prove that if $A$ is a $\\pi$-group of automorphisms of a $\\pi$-soluble finite group $G$ such that any subset of $I_G(A)$ generates a subgroup that can be generated by $r$ elements, then the rank of $[G,A]$ is bounded in terms of $r$. Examples show that such a result does not hold without the assumption of $\\pi$-solubility. Earlier we obtained this type of results for groups of coprime automorphisms and for Sylow $p$-subgroups of $p$-soluble groups.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2025-05-05T20:37:51.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "local-global generation property",
      "soluble groups",
      "commutators",
      "soluble finite group",
      "coprime automorphisms"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}