{
  "id": "1611.06546",
  "version": "v1",
  "published": "2016-11-20T17:04:03.000Z",
  "updated": "2016-11-20T17:04:03.000Z",
  "title": "A characterisation of elementary abelian 3-groups",
  "authors": [
    "Chimere Anabanti"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "Tarnauceanu [Archiv der Mathematik, 102 (1), (2014), 11--14] gave a characterisation of elementary abelian $2$-groups in terms of their maximal sum-free sets. His theorem states that a finite group $G$ is an elementary abelian $2$-group if and only if the set of maximal sum-free sets coincides with the set of complements of the maximal subgroups. A corollary is that the number of maximal sum-free sets in an elementary abelian $2$-group of finite rank $n$ is $2^n-1$. Regretfully, we show here that the theorem is wrong. We then prove a correct version of the theorem from which the desired corollary can be deduced. Moreover, we give a characterisation of elementary abelian $3$-groups in terms of their maximal sum-free sets. A corollary to our result is that the number of maximal sum-free sets in an elementary abelian $3$-group of finite rank $n$ is $3^n-1$. Finally, for prime $p>3$ and $n\\in \\mathbb{N}$, we show that there is no direct analogue of this result for elementary abelian $p$-groups of finite rank $n$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-11-20T17:04:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B75",
      "20D60",
      "20K01",
      "05E15"
    ],
    "keywords": [
      "elementary abelian",
      "finite rank",
      "characterisation",
      "maximal sum-free sets coincides",
      "archiv der mathematik"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}