Fuglede's conjecture holds on cyclic groups ${\mathbb{Z}}_{pqr}$

Fuglede’s spectral set conjecture states that a subset $\mathrm{\Omega }$ of a locally compact abelian group $G$ tiles the group by translation if and only if there exists a subset of continuous group characters which is an orthogonal basis of $L^2(\mathrm{\Omega })$. We prove that Fuglede’s conjecture holds on cyclic groups ${\mathbb{Z}}_{pqr}$ with $p,q,r$ distinct primes.