{
  "id": "2001.11106",
  "version": "v1",
  "published": "2020-01-29T21:47:31.000Z",
  "updated": "2020-01-29T21:47:31.000Z",
  "title": "The order of the product of two elements in finite nilpotent groups",
  "authors": [
    "C. M. Bonciocat"
  ],
  "comment": "20 pages",
  "categories": [
    "math.GR"
  ],
  "abstract": "An old problem in group theory is that of describing how the order of an element behaves under multiplication. To generalize some classical bounds concerning the order $\\mathrm o(ab)$ of two elements $a, b$ in a finite abelian group to the non-commutative case, we replace $\\mathrm o(ab)$ with a notion of mutual order $\\mathrm o(a, b)$, defined as the least positive integer $n$ such that $a^nb^n = 1$. Motivated by this, we then compare $\\mathrm o(ab)$ and $\\mathrm o(a, b)$ in finite nilpotent groups, and show that in a group of class $\\gamma$, the ratio $\\mathrm o(ab)/\\mathrm o(a, b)$ lies in some fixed finite set $S(\\gamma) \\subset \\mathbb Q$, whose elements do not involve prime factors exceeding $\\gamma$. In particular, we generalize a result of P. Hall, which asserts that $\\mathrm o(ab) = \\mathrm o(a, b)$ in $p$-groups with $p > \\gamma$. We end with a more detailed analysis for groups of class 2, which allows one to give a more explicit description of $\\mathrm o(ab)/\\mathrm o(a, b)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-01-29T21:47:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20D15",
      "20A05",
      "20F12",
      "20F50",
      "20F69",
      "11A07"
    ],
    "keywords": [
      "finite nilpotent groups",
      "finite abelian group",
      "explicit description",
      "group theory",
      "old problem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}