{
  "id": "math/0610983",
  "version": "v2",
  "published": "2006-10-31T19:03:42.000Z",
  "updated": "2007-12-25T13:40:22.000Z",
  "title": "A commutator description of the solvable radical of a finite group",
  "authors": [
    "Nikolai Gordeev",
    "Fritz Grunewald",
    "Boris Kunyavskii",
    "Eugene Plotkin"
  ],
  "comment": "43 pages",
  "journal": "Groups, Geometry, and Dynamics 2 (2008), No. 1, 85-120",
  "categories": [
    "math.GR"
  ],
  "abstract": "We are looking for the smallest integer k>1 providing the following characterization of the solvable radical R(G) of any finite group G: R(G) coincides with the collection of all g such that for any k elements a_1,a_2,...,a_k the subgroup generated by the elements g, a_iga_i^{-1}, i=1,...,k, is solvable. We consider a similar problem of finding the smallest integer l>1 with the property that R(G) coincides with the collection of all g such that for any l elements b_1,b_2,...,b_l the subgroup generated by the commutators [g,b_i], i=1,...,l, is solvable. Conjecturally, k=l=3. We prove that both k and l are at most 7. In particular, this means that a finite group G is solvable if and only if in each conjugacy class of G every 8 elements generate a solvable subgroup.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2007-12-25T13:40:22.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "finite group",
      "commutator description",
      "solvable radical",
      "smallest integer",
      "collection"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 43,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math.....10983G"
    }
  }
}