Abstract: In 1974, Fuglede conjectured that any bounded set $\Omega$ in $\Bbb R^d$, with positive measure, tiles $\Bbb R^d$ by countable many translations if and only if $L^2(\Omega)$ has an orthogonal exponential basis. The conjecture was disproved by T. Tao in 2003 in dimensions $d=5$ and higher, followed by results of other people, where the conjecture was disproved in dimension 3 and 4. In this talk we shall look at the finite version of this conjecture and show that the conjecture holds in $\Bbb Z_p^2$, $p$ prime.