{
  "id": "2103.08353",
  "version": "v1",
  "published": "2021-03-15T12:48:48.000Z",
  "updated": "2021-03-15T12:48:48.000Z",
  "title": "Factorizations of groups of small order",
  "authors": [
    "Mikhail Kabenyuk"
  ],
  "comment": "13 pages",
  "categories": [
    "math.GR"
  ],
  "abstract": "Let $G$ be a finite group and let $A_1,\\ldots,A_k$ be a collection of subsets of $G$ such that $G=A_1\\ldots A_k$ is the product of all the $A_i$'s with $|G|=|A_1|\\ldots|A_k|$. We write $G=A_1\\cdot\\ldots\\cdot A_k$ and call this a $k$-fold factorization of $G$ of the form $(|A_1|,\\ldots,|A_k|)$ or more briefly an $(|A_1|,\\ldots,|A_k|)$-factorization of $G$. Let $k\\geq2$ be a fixed integer. If $G$ has an $(a_1,\\ldots,a_k)$-factorization, whenever $|G|=a_1\\ldots a_k$ with $a_i>1$, $i=1,\\ldots,k$, we say that $G$ is $k$-factorizable. We say that $G$ is multifold-factorizable if $G$ is $k$-factorizable for any possible integer $k\\geq2$. In this paper we prove that there are exactly $6$ non-multifold-factorizable groups among the groups of order at most $60$. Here is their complete list: $A_4$, $(C_2\\times C_2)\\rtimes C_9$, $A_4\\times C_3$, $(C_2\\times C_2\\times C_2)\\rtimes C_7$, $A_5$, $A_4\\times C_5$. Some related open questions are presented.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-03-15T12:48:48.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20D60"
    ],
    "keywords": [
      "small order",
      "fold factorization",
      "finite group",
      "complete list",
      "related open questions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}