{
  "id": "2111.05394",
  "version": "v3",
  "published": "2021-11-09T20:04:46.000Z",
  "updated": "2022-03-15T17:47:54.000Z",
  "title": "Zero-sum partitions of Abelian groups of order $2^n$",
  "authors": [
    "Sylwia Cichacz",
    "Karol Suchan"
  ],
  "categories": [
    "math.CO",
    "math.GR"
  ],
  "abstract": "The following problem has been known since the 80's. Let $\\Gamma$ be an Abelian group of order $m$ (denoted $|\\Gamma|=m$), and let $t$ and $m_i$, $1 \\leq i \\leq t$, be positive integers such that $\\sum_{i=1}^t m_i=m-1$. Determine when $\\Gamma^*=\\Gamma\\setminus\\{0\\}$, the set of non-zero elements of $\\Gamma$, can be partitioned into disjoint subsets $S_i$, $1 \\leq i \\leq t$, such that $|S_i|=m_i$ and $\\sum_{s\\in S_i}s=0$ for every $i$, $1 \\leq i \\leq t$. It is easy to check that $m_i\\geq 2$ (for every $i$, $1 \\leq i \\leq t$) and $|I(\\Gamma)|\\neq 1$ are necessary conditions for the existence of such partitions, where $I(\\Gamma)$ is the set of involutions of $\\Gamma$. It was proved that the condition $m_i\\geq 2$ is sufficient if and only if $|I(\\Gamma)|\\in\\{0,3\\}$. For other groups (i.e., for which $|I(\\Gamma)|\\neq 3$ and $|I(\\Gamma)|>1$), only the case of any group $\\Gamma$ with $\\Gamma\\cong(Z_2)^n$ for some positive integer $n$ has been analyzed completely so far, and it was shown independently by several authors that $m_i\\geq 3$ is sufficient in this case. Moreover, recently Cichacz and Tuza proved that, if $|\\Gamma|$ is large enough and $|I(\\Gamma)|>1$, then $m_i\\geq 4$ is sufficient. In this paper we generalize this result for every Abelian group of order $2^n$. Namely, we show that the condition $m_i\\geq 3$ is sufficient for $\\Gamma$ such that $|I(\\Gamma)|>1$ and $|\\Gamma|=2^n$, for every positive integer $n$. We also present some applications of this result to graph magic- and anti-magic-type labelings.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2022-03-15T17:47:54.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E16",
      "20K01",
      "05C25",
      "05C78",
      "G.2.1",
      "G.2.2"
    ],
    "keywords": [
      "abelian group",
      "zero-sum partitions",
      "positive integer",
      "sufficient",
      "non-zero elements"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}