Automorphisms of the doubles of purely non-abelian finite groups

Abstract

Using a recent classification of $\mathrm{End}(\mathcal{D}(G))$, we determine a number of properties for $\mathrm{Aut}(\mathcal{D}(G))$, where $\mathcal{D}(G)$ is the Drinfel'd double of a finite group $G$. Furthermore, we completely describe $\mathrm{Aut}(\mathcal{D}(G))$ for all purely non-abelian finite groups $G$. A description of the action of $\mathrm{Aut}(\mathcal{D}(G))$ on $\mathrm{Rep}(\mathcal{D}(G))$ is also given. We are also able to produce a simple proof that $\mathcal{D}(G)\cong \mathcal{D}(H)$ if and only if $G\cong H$, for $G$ and $H$ finite groups.

arXiv comment39 pages; title changed; results expanded; notation and proofs substantially improved; a few corrections (proof that u was always Hopf was flawed, and has been removed)