Grand unification and proton stability near the Peccei-Quinn scale

We show that in an SU(2) ⊗ U(1) model with a Dine-Fischler- Srednicki-like invisible axion it is possible to obtain (i) the convergence of the three gauge coupling constants at an energy scale near the Peccei-Quinn scale; (ii) the correct value for sin<sup>2</sup>θ<inf>W</inf>(M <inf>Z</inf>); (iii) the stabilization of the proton by the cyclic Z <inf>13</inf> ⊗ Z<inf>3</inf> symmetries which also stabilize the axion as a solution to the strong CP problem. Concerning the convergence of the three coupling constants and the prediction of the weak mixing angle at the Z peak, this model is as good as the minimal supersymmetric standard model with μ<inf>SUSY</inf> = M<inf>Z</inf>. We also consider the standard model with six and seven Higgs doublets. The main calculations were done in the 1-loop approximation but we briefly consider the 2-loop contributions.