Abstract: Mutually unbiased bases in Hilbert spaces of finite dimensions are closely related to the quantal notion of complementarity. An alternative proof of existence of a maximal collection of $N+1$ mutually unbiased bases in Hilbert spaces of prime dimension $N$ is given by exploiting the finite Heisenberg group (also called the Pauli group) and the action of $SL(2,\mathbb{Z}_N)$ on finite phase space $\mathbb{Z}_N \times \mathbb{Z}_N$ implemented by unitary operators in the Hilbert space. Crucial for the proof is that, for prime $N$, $\mathbb{Z}_N$ is also a finite field.