{
  "id": "2006.03786",
  "version": "v1",
  "published": "2020-06-06T05:32:22.000Z",
  "updated": "2020-06-06T05:32:22.000Z",
  "title": "Transversals, near transversals, and diagonals in iterated groups and quasigroups",
  "authors": [
    "Anna A. Taranenko"
  ],
  "comment": "17 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "Given a binary quasigroup $G$ of order $n$, a $d$-iterated quasigroup $G[d]$ is the $(d+1)$-ary quasigroup equal to the $d$-times composition of $G$ with itself. The Cayley table of every $d$-ary quasigroup is a $d$-dimensional latin hypercube. Transversals and diagonals in multiary quasigroups are defined so that to coincide with those in the corresponding latin hypercube. We prove that if a group $G$ of order $n$ satisfies the Hall--Paige condition, then the number of transversals in $G[d]$ is equal to $ \\frac{n!}{ |G'| n^{n-1}} \\cdot n!^{d} (1 + o(1))$ for large $d$, where $G'$ is the commutator subgroup of $G$. For a general quasigroup $G$, we obtain similar estimations on the numbers of transversals and near transversals in $G[d]$ and develop a method for counting diagonals of other types in iterated quasigroups.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-06-06T05:32:22.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "transversals",
      "iterated groups",
      "iterated quasigroup",
      "ary quasigroup equal",
      "dimensional latin hypercube"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}