{
  "id": "1409.7903",
  "version": "v1",
  "published": "2014-09-28T13:01:29.000Z",
  "updated": "2014-09-28T13:01:29.000Z",
  "title": "The Answers to a Problem and Two Conjectures about OD-Characterization of Finite Groups",
  "authors": [
    "Ali Mahmoudifar",
    "Behrooz Khosravi"
  ],
  "comment": "3 pages",
  "categories": [
    "math.GR"
  ],
  "abstract": "In [Akbari and Moghaddamfar, Recognizing by order and degree pattern of some projective special linear groups, {\\it Internat. J. Algebra Comput.}, 2012] the authors possed the following problem: \\\\ {\\bf Problem.} {\\it Is there a simple group which is $k$-fold OD-characterizable for $k\\geq3\\ ?$ } In this paper as the main result we give positive answer to the above problem and we introduce two simple groups which are $k$-fold OD-characterizable such that $k\\geq6$. Also in [R. Kogani-Moghadam and A. R. Moghaddamfar, Groups with the same order and degree pattern, {\\it Science China Mathematics}, 2012], the authors possed two conjectures as follows: \\\\ {\\bf Conjecture 1.} {\\it All alternating groups $A_m$ with $m \\not= 10$ are OD-characterizable.} \\\\ {\\bf Conjecture 2.} {\\it All symmetric groups $S_m$, with $m \\not= 10$, are $n$-fold OD-characterizable, where $n\\in\\{1, 3\\}$.} In this paper we find some alternating and some symmetric groups such that these conjectures are not true for them.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-09-28T13:01:29.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "finite groups",
      "conjecture",
      "symmetric groups",
      "simple group",
      "fold od-characterizable"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 3,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1409.7903M"
    }
  }
}