\begin{document}$SU(5)$\end{document} and \begin{document}$ SU(6) $\end{document} to determine possible grand unified theories that give rise to three-generational Standard Model fermions without simple repetitions. By demanding asymptotic free theories at short distances, we find gauge groups of \begin{document}$ SU(7) $\end{document}, \begin{document}$ SU(8) $\end{document}, and \begin{document}$ SU(9) $\end{document}, together with their anomaly-free irreducible representations, are such candidates. Two additional gauge groups of \begin{document}$ SU(10) $\end{document} and \begin{document}$ SU(11) $\end{document} can also achieve the generational structure without asymptotic freedom. We also deduce that these models can solve the Peccei-Quinn (PQ) quality problem, which is intrinsic in the axion models, with the leading PQ-breaking operators determined from the symmetry requirement."> High-quality grand unified theories with three generations -
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