We show that for each natural p >= 2, the Lefschetz fixed point theorem is optimal when applied to Z(p)-actions by homeomorphisms on the three dimensional torus T-3. More precisely, we show that for a spectrally unitary Z(p)-action A on the first homology group H-1 (T-3, Z) with trivial fixed point set, there exists a free Zr-action by real analytic diffeomorphisms of T-3 whose induced Z(p)-action on H-1(T-3, Z) is the action A. In particular, we establish the normal form for this type of actions.