{
  "id": "1011.2503",
  "version": "v3",
  "published": "2010-11-10T21:07:31.000Z",
  "updated": "2011-12-19T07:40:21.000Z",
  "title": "A new subgroup lattice characterization of finite solvable groups",
  "authors": [
    "John Shareshian",
    "Russ Woodroofe"
  ],
  "comment": "15 pages; v2 has minor changes for publication; v3 minor typos fixed",
  "journal": "J. Algebra 351 (2012), no. 1, 448-458",
  "doi": "10.1016/j.jalgebra.2011.10.032",
  "categories": [
    "math.GR",
    "math.CO"
  ],
  "abstract": "We show that if G is a finite group then no chain of modular elements in its subgroup lattice L(G) is longer than a chief series. Also, we show that if G is a nonsolvable finite group then every maximal chain in L(G) has length at least two more than that of the chief length of G, thereby providing a converse of a result of J. Kohler. Our results enable us to give a new characterization of finite solvable groups involving only the combinatorics of subgroup lattices. Namely, a finite group G is solvable if and only if L(G) contains a maximal chain X and a chain M consisting entirely of modular elements, such that X and M have the same length.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2011-12-19T07:40:21.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20E15",
      "20D30",
      "06A07"
    ],
    "keywords": [
      "finite solvable groups",
      "subgroup lattice characterization",
      "modular elements",
      "maximal chain",
      "chief series"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1011.2503S"
    }
  }
}