Transversals, near transversals, and diagonals in iterated groups and quasigroups

Anna A. Taranenko¹

¹Sobolev Institute of Mathematics

The electronic journal of combinatorics, Tome 28 (2021) no. 3

Résumé

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 as 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.

Détail

DOI : 10.37236/9699

Classification : 05B15, 05D15, 05A16, 05E16, 20N05
Mots-clés : binary quasigroup, Cayley table