On the exponent of tensor categories coming from finite groups

Abstract

We describe the exponent of a group-theoretical fusion category $\mathcal{C}=\mathcal{C}(G,\omega ,F,\alpha )$ associated to a finite group $G$ in terms of group cohomology. We show that the exponent of $\C$ divides both $e(\omega )\exp G$ and $(\exp G{)}^2$, where $e(\omega )$ is the cohomological order of the 3-cocycle $\omega$. In particular $\exp \C$ divides $(\dim \C)^2$.

22 pages, amslatex