Universal Spectra in \(G\times {\mathbb {Z}}_p\)

Let G be an additive and finite Abelian group, and p a prime number that does not divide the order of G. We show that if G has the universal spectrum property, then so does \(G\times {\mathbb {Z}}_p\) $G\times {\mathbb{Z}}_p$ . This is similar to and extends a previous result for cyclic groups using the same dilation trick but on non-cyclic groups as well. Inductively applying this statement on the known list of permissible G one can replace p with any square-free number that does not divide the order of G, and produce new tiling to spectral results in finite Abelian groups generated by at most two elements.