{
  "id": "1704.02471",
  "version": "v1",
  "published": "2017-04-08T10:35:38.000Z",
  "updated": "2017-04-08T10:35:38.000Z",
  "title": "Difference bases in finite Abelian groups",
  "authors": [
    "Taras Banakh",
    "Volodymyr Gavrylkiv"
  ],
  "comment": "12 pages",
  "categories": [
    "math.CO",
    "math.GR"
  ],
  "abstract": "A subset $B$ of a group $G$ is called a difference basis of $G$ if each element $g\\in G$ can be written as the difference $g=ab^{-1}$ of some elements $a,b\\in B$. The smallest cardinality $|B|$ of a difference basis $B\\subset G$ is called the difference size of $G$ and is denoted by $\\Delta[G]$. The fraction $\\eth[G]:=\\frac{\\Delta[G]}{\\sqrt{|G|}}$ is called the difference characteristic of $G$. Using properies of the Galois rings, we prove recursive upper bounds for the difference sizes and characteristics of finite Abelian groups. In particular, we prove that for a prime number $p\\ge 11$, any finite Abelian $p$-group $G$ has difference characteristic $\\eth[G]<\\frac{\\sqrt{p}-1}{\\sqrt{p}-3}\\cdot\\sup_{k\\in\\mathbb N}\\eth[C_{p^k}]<\\sqrt{2}\\cdot\\frac{\\sqrt{p}-1}{\\sqrt{p}-3}$. Also we calculate the difference sizes of all Abelian groups of cardinality $<96$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-04-08T10:35:38.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05B10",
      "05E15",
      "16L99",
      "16Z99",
      "20D60",
      "20K01"
    ],
    "keywords": [
      "finite abelian groups",
      "difference basis",
      "difference size",
      "difference characteristic",
      "prime number"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}