Transversals, Near Transversals, and Diagonals in Iterated Groups and Quasigroups

Given a binary quasigroup $G$ of order $n$, $d$-iterated $G[d]$ is the $(d+1)$-ary equal to $d$-times composition with itself. The Cayley table every $d$-ary $d$-dimensional latin hypercube. Transversals and diagonals in multiary quasigroups are defined so as coincide those corresponding hypercube.
 We prove that if group $n$ satisfies Hall–Paige condition, then number transversals $ \frac{n!}{ |G'| n^{n-1}} \cdot n!^{d} (1 + o(1))$ for large $d$, where $G'$ commutator subgroup $G$. For general $G$, we obtain similar estimations on numbers near develop method counting other types iterated quasigroups.