Anomalous Discrete Symmetries in Three Dimensions and Group Cohomology

We study ’t Hooft anomalies for a global discrete internal symmetry $G$. We construct examples of bosonic field theories in three dimensions with a nonvanishing ’t Hooft anomaly for a discrete global symmetry. We also construct field theories in three dimensions with a global discrete internal symmetry $G_1\times G_2$ such that gauging $G_1$ necessarily breaks $G_2$ and vice versa. This is analogous to the Adler-Bell-Jackiw axial anomaly in four dimensions and parity anomaly in three dimensions.