Abstract
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), $\mathrm{SU}(4)\ifmmode\times\else\texttimes\fi{}{\mathrm{SU}(2)}_{L}\ifmmode\times\else\texttimes\fi{}{\mathrm{SU}(2)}_{R}$, ${\mathrm{SU}(3)}_{C}\ifmmode\times\else\texttimes\fi{}{\mathrm{SU}(3)}_{L}\ifmmode\times\else\texttimes\fi{}{\mathrm{SU}(3)}_{R}$, and SU(6)\ifmmode\times\else\texttimes\fi{}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.
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