The pairing symmetry, mathematically defined as the representation of the unbroken symmetry group carried by the pairing order parameter, is considered as a classification and characterization of the broken symmetries in a superconductor. In this talk, we will introduce another concept, known as the projective symmetry group (PSG) of a superconductor. We will show that the PSG imposes a strong constraint on the pairing symmetry, and in many cases has a one-to-one correspondence with the pairing symmetry. This map from PSG to pairing symmetry allows us to identify two new ways to detect pairing symmetry in a superconductor: (1) through new selection rules in optical/THz and Raman spectroscopy, (2) through the presence/absence of Majorana zero modes in vortex cores centered at high-symmetry locations of the crystal. I will illustrate the results using examples of superconductors w/ or w/o spin-orbit interactions.