I will discuss how symmetries of quantum spin systems can be realized. For a given realization of a symmetry group G of a 1d spin system, I will define the anomalous index that takes values in the cohomology H^4(BG) of the classifying space of the group. I will show that a G-invariant system with a non-trivial anomalous index can not have a gapped G-invariant ground state, that provides a generalization of Lieb-Schultz-Mattis theorem when the translation symmetry is not necessarily present.