{
  "id": "1310.1643",
  "version": "v5",
  "published": "2013-10-06T22:56:18.000Z",
  "updated": "2014-12-12T16:47:18.000Z",
  "title": "On the Erdos-Ko-Rado property for finite Groups",
  "authors": [
    "Mohammad Bardestani",
    "Keivan Mallahi-Karai"
  ],
  "comment": "This is the final version. To appear in the Journal of Algebraic Combinatorics",
  "categories": [
    "math.GR",
    "math.CO"
  ],
  "abstract": "Let a finite group $G$ act transitively on a finite set $X$. A subset $S\\subseteq G$ is said to be {\\it intersecting} if for any $s_1,s_2\\in S$, the element $s_1^{-1}s_2$ has a fixed point. The action is said to have the {\\it weak Erd\\H{o}s-Ko-Rado} property, if the cardinality of any intersecting set is at most $|G|/|X|$. If, moreover, any maximal intersecting set is a coset of a point stabilizer, the action is said to have the {\\it strong Erd\\H{o}s-Ko-Rado} property. In this paper we will investigate the weak and strong Erd\\H{o}s-Ko-Rado property and attempt to classify the groups whose all transitive actions have these properties. In particular, we show that a group with the weak Erd\\H{o}s-Ko-Rado property is solvable and that a nilpotent group with the strong Erd\\H{o}s-Ko-Rado property is product of a $2$-group and an abelian group of odd order.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2014-01-09T22:35:45.000Z",
      "comment": "Paper is reorganized. Theorem 1 is improved and a new corrected proof is given",
      "journal": null,
      "doi": null
    },
    {
      "version": "v5",
      "updated": "2014-12-12T16:47:18.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "finite group",
      "erdos-ko-rado property",
      "finite set",
      "maximal intersecting set",
      "point stabilizer"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1310.1643B"
    }
  }
}