Elements of Finite Order in Lie Groups and Discrete Gauge Symmetries

We apply Kac's theory of elements of finite order (EFO) in Lie groups to the description of discrete gauge symmetries in various supersymmetric grand unified models. Taking into account the discrete anomaly cancellation conditions, we identify the EFO which generate certain matter parities in the context of the supersymmetric $SO(10)$ and $E_6$ models.