{
  "id": "1101.4715",
  "version": "v3",
  "published": "2011-01-25T04:12:31.000Z",
  "updated": "2011-03-30T08:02:51.000Z",
  "title": "Extremal incomplete sets in finite abelian groups",
  "authors": [
    "Dan Guo",
    "Yongke Qu",
    "Guoqing Wang",
    "Qinghong Wang"
  ],
  "comment": "Swith some notations into ones which are more popular and put the materials of Appendix into the body. The present version is to appear in Ars Combinatoria",
  "categories": [
    "math.CO",
    "math.NT"
  ],
  "abstract": "Let $G$ be a finite abelian group. The critical number ${\\rm cr}(G)$ of $G$ is the least positive integer $\\ell$ such that every subset $A\\subseteq G\\setminus\\{0\\}$ of cardinality at least $\\ell$ spans $G$, i.e., every element of $G$ can be written as a nonempty sum of distinct elements of $A$. The exact values of the critical number have been completely determined recently for all finite abelian groups. The structure of these sets of cardinality ${\\rm cr}(G)-1$ which fail to span $G$ has also been characterized except for the case that $|G|$ is an even number and the case that $|G|=pq$ with $p,q$ are primes. In this paper, we characterize these extremal subsets for $|G|\\geq 36$ is an even number, or $|G|=pq$ with $p,q$ are primes and $q\\geq 2p+3$.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2011-03-30T08:02:51.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "finite abelian group",
      "extremal incomplete sets",
      "critical number",
      "extremal subsets",
      "exact values"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1101.4715G"
    }
  }
}