{
  "id": "1105.0475",
  "version": "v1",
  "published": "2011-05-03T04:20:51.000Z",
  "updated": "2011-05-03T04:20:51.000Z",
  "title": "A new solvability criterion for finite groups",
  "authors": [
    "Silvio Dolfi",
    "Robert Guralnick",
    "Marcel Herzog",
    "Cheryl Praeger"
  ],
  "comment": "The results here are an improved version of the paper of the same name posted as arXiv:1007.5394 by the first, third and fourth authors",
  "doi": "10.1112/jlms/jdr041",
  "categories": [
    "math.GR"
  ],
  "abstract": "In 1968, John Thompson proved that a finite group G is solvable if and only if every 2-generator subgroup of G is solvable. In this paper, we prove that solvability of a finite group G is guaranteed by a seemingly weaker condition: G is solvable if, for all conjugacy classes C and D of G consisting of elements of prime power order, there exist x in C and y in D with x and y generating a solvable group. We also prove the following property of finite nonabelian simple groups, which is the key tool for our proof of the solvability criterion: if G is a finite nonabelian simple group, then there exist two prime divisors a and b of |G| such that, for all elements x, y in G with |x|=a and |y|=b, the subgroup generated by x and y is not solvable. Further, using a recent result of Guralnick and Malle, we obtain a similar membership criterion for any family of finite groups closed under forming subgroups, quotients and extensions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-05-03T04:20:51.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20D10",
      "20F16"
    ],
    "keywords": [
      "finite group",
      "solvability criterion",
      "finite nonabelian simple group",
      "prime power order",
      "similar membership criterion"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1105.0475D"
    }
  }
}