Discrete gauge symmetries in supersymmetric grand unified models

We investigate the presence of discrete gauge symmetries in grand unification models based on SO(10) and ${\mathrm{E}}_6$. These models include flipped and unflipped SU(5), $\mathrm{SU}\mathrm{}\left(4\right)\mathrm{}\times {\mathrm{SU}\mathrm{}\left(2\right)\mathrm{}}_L\times {\mathrm{SU}\mathrm{}\left(2\right)\mathrm{}}_R$, ${\mathrm{SU}\mathrm{}\left(3\right)\mathrm{}}_C\times {\mathrm{SU}\mathrm{}\left(3\right)\mathrm{}}_L\times {\mathrm{SU}\mathrm{}\left(3\right)\mathrm{}}_R$, and SU(6)×SU(2). Using the Dynkin formalism we find the U(1) subalgebras contained in the unified groups, give an expression for the Higgs fields that preserves each discrete symmetry, and determine the low-energy matter content implied by chirality. We discuss two $Z_2$ and three $Z_3$ non-equivalent cases. Among the possibilities found, only the usual $Z_2$ matter parity ($R$ parity) of supersymmetric extensions is consistent with a minimal matter content with no right-handed neutrinos, extra Higgs doublets, or nonstandard down-type quarks.