{
  "id": "1109.6559",
  "version": "v2",
  "published": "2011-09-29T15:29:06.000Z",
  "updated": "2011-12-21T16:02:00.000Z",
  "title": "Coprime subdegrees for primitive permutation groups and completely reducible linear groups",
  "authors": [
    "Silvio Dolfi",
    "Robert Guralnick",
    "Cheryl Praeger",
    "Pablo Spiga"
  ],
  "comment": "20 pages, 1 table",
  "categories": [
    "math.GR"
  ],
  "abstract": "In this paper we answer a question of Gabriel Navarro about orbit sizes of a finite linear group H acting completely reducibly on a vector space V: if the orbits containing the vectors a and b have coprime lengths m and n, we prove that the orbit containing a+b has length mn. Such groups H are always reducible if n and m are greater than 1. In fact, if H is an irreducible linear group, we show that, for every pair of non-zero vectors, their orbit lengths have a non-trivial common factor. In the more general context of finite primitive permutation groups G, we show that coprime non-identity subdegrees are possible if and only if G is of O'Nan-Scott type AS, PA or TW. In a forthcoming paper we will show that, for a finite primitive permutation group, a set of pairwise coprime subdegrees has size at most 2. Finally, as an application of our results, we prove that a field has at most 2 finite extensions of pairwise coprime indices with the same normal closure.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2011-12-21T16:02:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20H30"
    ],
    "keywords": [
      "reducible linear groups",
      "coprime subdegrees",
      "finite primitive permutation group",
      "non-trivial common factor",
      "finite linear group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1109.6559D"
    }
  }
}