{
  "id": "1510.03733",
  "version": "v1",
  "published": "2015-10-13T15:15:43.000Z",
  "updated": "2015-10-13T15:15:43.000Z",
  "title": "A finiteness condition on centralizers in locally finite groups",
  "authors": [
    "Gustavo A. Fernandez-Alcober",
    "Leire Legarreta",
    "Antonio Tortora",
    "Maria Tota"
  ],
  "comment": "11 pages, submitted to an international journal. arXiv admin note: substantial text overlap with arXiv:1412.7330",
  "categories": [
    "math.GR"
  ],
  "abstract": "We consider a finiteness condition on centralizers in a group G, namely that |C_G (x) : <x>| is finite for every non-normal cyclic subgroup <x> of G. For periodic groups, this is the same as |C_G (x)| is finite for every non-normal cyclic subgroup <x> of G. We give a full description of locally finite groups satisfying this condition. As it turns out, they are a special type of cyclic extensions of Dedekind groups. We also study a variation of our condition, where the requirement of finiteness is replaced with a bound: |C_G (x) : <x>| < n for every non-normal cyclic subgroup <x> of G, for some fixed n. In this case, we are able to extend our analysis to the class of periodic locally graded groups.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-10-13T15:15:43.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F50",
      "20E34"
    ],
    "keywords": [
      "locally finite groups",
      "finiteness condition",
      "non-normal cyclic subgroup",
      "centralizers",
      "periodic groups"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv151003733F"
    }
  }
}