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Let $\mathcal{C}$ be a symmetric monoidal category generated by one element $X$ such that $End(X)=G$ where $G$ is a finite group. Is it true that, for any object $A \in \mathcal{C}$, $End(A)$ is isomorphic to a wreath product $G \wr S_n$, $n \in \mathbb{N}$ ?

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No. Take the category whose objects are vector spaces of dimension $2^n$ over $\mathbb{F}_2$ for $n \in \mathbb{Z}_{\ge 0}$ and whose morphisms are all isomorphisms. The object $X = \mathbb{F}_2^2$ generates this category under tensor product (I assume this is what you meant) and $\text{End}(X) \cong \text{GL}_2(\mathbb{F}_2) \cong S_3$. It should be straightforward to verify that $\text{End}(X^{\otimes n}) \cong \text{GL}_{2^n}(\mathbb{F}_2)$ has order larger than $S_3 \wr S_n$ for sufficiently large $n$.

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    @Alex: oh. In that case, I'm pretty sure what you want is true.2012-06-20