  Based on the idea of the Abelian group theory in mathematics,this paper finds
a sufficient condition of having independent TE and TM modes in a waveguide
filled with homogenous anisotropic lossless medium. For independent TE modes,
we prove the nonzero cut-off wavenumbers obtained from longitudinal scalar
magnetic field stimulation and transverse vector electric field stimulation are
same in theory. For independent TM modes, we also prove the nonzero cut-off
wavenumbers obtained from longitudinal scalar electric field stimulation and
transverse vector magnetic field stimulation are same in theory. Finally we
carry out several numerical experiments to verify the correctness of the
condition given by us.We hope that this condition is useful for the designs of
waveguide with homogenous anisotropic lossless medium in microwave engineering
community.
