{
  "id": "1412.7330",
  "version": "v1",
  "published": "2014-12-23T11:52:49.000Z",
  "updated": "2014-12-23T11:52:49.000Z",
  "title": "Some finiteness conditions on normalizers or centralizers in groups",
  "authors": [
    "Gustavo A. Fernandez-Alcober",
    "Leire Legarreta",
    "Antonio Tortora",
    "Maria Tota"
  ],
  "categories": [
    "math.GR"
  ],
  "abstract": "We consider several finiteness conditions on normalizers and centralizers in a group G, the most important of which are the following two: (i) the index |N_G(H):H| is finite for every non-normal subgroup H of G, and (ii) the index |C_G(x):<x>| is finite for every non-normal cyclic subgroup <x> of G. We show that (i) and (ii) are equivalent in the realms of locally finite groups and locally nilpotent groups, in which case we can give a full description of the groups satisfying these conditions. As it turns out, they are a special type of cyclic extensions of Dedekind groups. We also study the following variations of conditions (i) and (ii), where the requirement of finiteness is replaced with a bound: (i') |N_G(H):H| <= n for every non-normal subgroup H of G, and (ii') |C_G(x):<x>| <= n for every non-normal cyclic subgroup <x> of G, for some fixed n. These conditions are equivalent to (i) and (ii) for locally finite groups, but on the contrary, they imply that the group G is abelian for non-periodic locally nilpotent groups. Also, we are able to extend our analysis of conditions (i') and (ii') to the classes of non-periodic groups and periodic locally graded groups.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-12-23T11:52:49.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "finiteness conditions",
      "centralizers",
      "normalizers",
      "locally finite groups",
      "non-normal cyclic subgroup"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1412.7330F"
    }
  }
}