  Fuglede's conjecture states that a subset $\Omega\subseteq\mathbb{R}^{n}$ of
positive and finite Lebesgue measure is a spectral set if and only if it tiles
$\mathbb{R}^{n}$ by translation. The conjecture does not hold in both
directions for $\mathbb{R}^n$, $n\ge3$. However, this conjecture remains open
in $\mathbb{R}$ and $\mathbb{R}^2$. Cyclic groups play important roles in the
study of Fuglede's conjecture in $\mathbb{R}$. In this paper, we introduce a
new tool to study the spectral sets in cyclic groups. In particular, we prove
that Fuglede's conjecture holds in $\mathbb{Z}_{p^{n}qr}$.
