Product groups, discrete symmetries, and grand unification

We study grand unified theories (GUT) based on an $\mathrm{SU}\left(5\right)\mathrm{}\times \mathrm{SU}\left(5\right)\mathrm{}$ gauge group in which the GUT scale, $M_{\mathrm{GUT}},$ is the vacuum expectation value of an exact or approximate modulus, and in which fast proton decay is avoided through a combination of a large triplet mass and small triplet couplings. These features are achieved by discrete symmetries. In many of our models, $M_{\mathrm{GUT}}$ is generated naturally by the balance of higher dimension terms that lift the GUT modulus potential, and soft supersymmetry breaking masses. The theories often lead to interesting patterns of quark and lepton masses. We also discuss some distinctions between grand unified theories and string unification.