On gauging finite subgroups

We study in general spacetime dimension the symmetry of the theory obtained by gauging a non-anomalous finite normal Abelian subgroup A A of a Γ -symmetric theory. Depending on how anomalous Γ is, we find that the symmetry of the gauged theory can be i) a direct product of G=/A G = Γ / A and a higher-form symmetry A A ̂ with a mixed anomaly, where A A ̂ is the Pontryagin dual of A A ; ii) an extension of the ordinary symmetry group G G by the higher-form symmetry A A ̂ ; iii) or even more esoteric types of symmetries which are no longer groups. We also discuss the relations to the effect called the H³ (G, A) H 3 (G, A ̂) symmetry localization obstruction in the condensed-matter theory and to some of the constructions in the works of Kapustin-Thorngren and Wang-Wen-Witten.