We present counterexamples to the lore that symmetries that cannot be gauged or made on site are necessarily anomalous. Specifically, we construct unitary, internal symmetries of two-dimensional lattice models that cannot be consistently coupled to background or dynamical gauge fields or disentangled to a tensor product of on-site operators. These symmetries are nevertheless anomaly-free in the sense that they admit symmetric, gapped Hamiltonians with unique, invertible ground states. We show that symmetries of this kind are characterized by an index $[\omega ]\in H^2(G,{\mathbb{Q}}_{+})$, where ${\mathbb{Q}}_{+}$ is the multiplicative group of positive rational numbers labeling one-dimensional quantum cellular automata.

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• Anomalies
• Symmetry protected topological states
• Discrete symmetries in condensed matter
• Spin lattice models