{
  "id": "1705.08685",
  "version": "v1",
  "published": "2017-05-24T10:14:07.000Z",
  "updated": "2017-05-24T10:14:07.000Z",
  "title": "The block graph of a finite group",
  "authors": [
    "Julian Brough",
    "Yanjun Liu",
    "Alessandro Paolini"
  ],
  "comment": "24 pages",
  "categories": [
    "math.RT",
    "math.GR"
  ],
  "abstract": "This paper studies intersections of principal blocks of a finite group with respect to different primes. We first define the block graph of a finite group G, whose vertices are the prime divisors of |G| and there is an edge between two vertices p \\ne q if and only if the principal p- and q-blocks of G have a nontrivial common complex irreducible character of G. Then we determine the block graphs of finite simple groups, which turn out to be complete except those of J_1 and J_4. Also, we determine exactly when the Steinberg character of a finite simple group of Lie type lies in a principal block. Based on the above investigation, we obtain a criterion for the p-solvability of a finite group which in particular asserts that a finite group whose block graph has no triangle containing 2 is solvable.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-05-24T10:14:07.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20C20",
      "20C33",
      "20D06"
    ],
    "keywords": [
      "finite group",
      "block graph",
      "finite simple group",
      "nontrivial common complex irreducible character",
      "principal block"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}