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

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 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.