  In this study, we consider one-dimension (1D) quantum spin systems with the
translation and discrete symmetries (spin reversal, space inversion and time
reversal symmetries). By combining the continuous U(1) symmetry with the
discrete symmetries and using the extended Lieb-Schultz-Mattis theorem
\cite{Lieb-Schultz-Mattis-1961}\cite{Nomura-Morishige-Isoyama-2015}, we
investigate the relation between the ground states, energy spectra and
symmetries. For half-integer spin cases, we generalize the dimer and N\'eel
concepts using the discrete symmetries, and we can reconcile the LSM theorem
with the dimer or N\'eel states, since there was a subtle dilemma. Furthermore,
a part of discrete symmetries is enough to classify possible phases. Thus we
can deepen our understanding of the relation between the LSM theorem and the
discrete symmetries.
