{
  "id": "2401.09306",
  "version": "v1",
  "published": "2024-01-17T16:17:43.000Z",
  "updated": "2024-01-17T16:17:43.000Z",
  "title": "Factorizations of simple groups of order 168 and 360",
  "authors": [
    "Mikhail Kabenyuk"
  ],
  "comment": "18 pages, any comments are welcome",
  "categories": [
    "math.GR"
  ],
  "abstract": "A finite group $G$ is called $k$-factorizable if for any factorization $|G|=a_1\\cdots a_k$ with $a_i>1$ there exist subsets $A_i$ of $G$ with $|A_i|=a_i$ such that $G=A_1\\cdots A_k$. We say that $G$ is \\textit{multifold-factorizable} if $G$ is $k$-factorizable for any possible integer $k\\geq2$. We prove that simple groups of orders 168 and 360 are multifold-factorizable and formulate two conjectures that the symmetric group $S_n$ for any $n$ and the alternative group $A_n$ for $n\\geq6$ are multifold-factorizable.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-01-17T16:17:43.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20B30",
      "20D40"
    ],
    "keywords": [
      "simple groups",
      "factorization",
      "finite group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}