Finite SU(N)k unification -: art. no. 026

We consider N = 1 supersymmetric gauge theories based on the group SU (N)(1) x SU (N)(2) x (...) x SU (N)(k) with matter content (N, N*, 1,..., 1) + ( 1, N, N*,..., 1) + (...) + ( N*, 1, 1,..., N) as candidates for the unifcationsymmetry of all particles. In particular we examine to which extent such theories can become finite and we find that a necessary condition is that there should be exactly three families. We discuss further some phenomenologic issues related to the cases (N; k) = (3; 3), (3; 4), and (4; 3), in an attempt to choose those theories that can become also realistic. Thus we are naturally led to consider the SU(3)(3) model which we first promote to an all-loop finite theory and then we study its additional predictions concerning the top quark mass, Higgs mass and supersymmetric spectrum.