Factorization of Finite Cyclic Group $\Bbb Z_{(pqr)^2}$: Szab\'{o} Pairs and Full Tiling Structures

In the study of factorizations of finite cyclic groups, a classical problem is to investigate the properties of factorization sets $A$ and $B$ in the direct sum decomposition $A \oplus B = \mathbb{Z}_{M}$ with $|A| = |B| =\sqrt{M}$, where $M=(pqr)^2$ for some distinct primes $p$, $q$, and $r$. In this paper, we show that neither $A$ nor $B$ is contained in a proper subgroup of $\mathbb{Z}_{(pqr)^2}$ if and only if the factorization sets $A, B$ form a Szab\'{o} pair. The factorization of finite cyclic groups is closely connected to the properties of tiling and spectral sets in $\Bbb Z$. The problem considered in this paper is equivalent to the simplest form of tiling that cannot be reduced to the two--prime case by the method provided by Coven and Meyerowitz (J. Algebra 212: 161--174, 1999). In contrast, the construction for the tiling which can be reduced to the two--prime case is already known. Our results present full structures for the factorization sets $A$ and $B$, and therefore, for this class of tilings.