Quasi-periodicity of \(\mathbb {Z}_{p^an_0}\)

Let p^a be a prime power and n₀ a square-free number. We prove that any complementing pair in a cyclic group of order p^an₀ is quasi-periodic, with one component decomposable by the the subgroup of order p. The proof is by induction and reduction since the presence of the square-free factor n₀ allows us to perform a Tijdeman decomposition. We also give an explicit example to show that \(\mathbb{Z}_{72}\) is the smallest cyclic group that fails to have the strong Tijdeman property.