Anomaly of non-Abelian discrete symmetries

We study anomalies of non-Abelian discrete symmetries; which part of non-Abelian group is anomaly free and which part can be anomalous. It is found that the anomaly-free elements of the group G generate a normal subgroup G(0) of G and the residue class group G/G(0), which becomes the anomalous part of G, is isomorphic to a single cyclic group. The derived subgroup D(G) of G is useful to study the anomaly structure. This structure also constrains the structure of the anomaly-free subgroup; the derived subgroup D(G) should be included in the anomaly-free subgroup. We study the detail structure of the anomaly-free subgroup from the structure of the derived subgroup in various discrete groups. For example, when G = S-n similar or equal to A(n) (sic) Z(2) and G = Delta (6n(2)) similar or equal to Delta(3n(2)) (sic) Z(2), in particular, A(n) and Delta (3n(2)) are at least included in the anomaly-free subgroup, respectively. This result holds in any arbitrary representations.

Automated summary

We study anomalies of non-Abelian discrete symmetries and find that the anomaly-free elements of the group generate a normal subgroup, while the residue class group becomes the anomalous part of the group. The derived subgroup is useful in studying the anomaly structure and constrains the structure of the anomaly-free subgroup.