Factorization of Finite Cyclic Group ${\mathbb{Z}}_{(pqr{)}^2}$: Szabó 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 $\mathrm{\mid }A\mathrm{\mid }=\mathrm{\mid }B\mathrm{\mid }=\sqrt{M}$, 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 Szabó pair. The factorization of finite cyclic groups is closely connected to the properties of tiling and spectral sets in $\mathbb{Z}$. 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.