  In this paper, we show an equi-disctributed property in $2$-dimensional
finite abelian groups $\mathbb{Z}_{p^2}\times \mathbb{Z}_{p}$ where $p$ is a
prime number. By using this equi-disctributed property, we prove that Fuglede's
spectral set conjecture holds on groups $\mathbb{Z}_{p^2}\times
\mathbb{Z}_{p}$, namely, a set in $\mathbb{Z}_{p^2}\times \mathbb{Z}_{p}$ is a
spectral set if and only if it is a tile.
