Peccei-Quinn symmetries without domains

In general, the unbroken discrete subgroup of a Peccei-Quinn symmetry gives rise to a domain problem—the formation of unacceptable domain walls in the course of the evolution of the early universe. In this paper, we construct an SU(5) model in which the unbroken discrete subgroup of a Peccei-Quinn symmetry is a trivial $Z_1$. That means that there is no domain problem for this model. It is pointed out that in models with more than one global U(1) symmetry it is easy to achieve a solution to the domain problem.