ScalingStacks

Proof. Denote by $f$ the map of the lemma. By [Lus, §3.6] (cf also [BjBr, Proposition 8.3.3]), the restriction of $f$ to $W_{n}$ induces an isomorphism with the subgroup of $\hat{{\mathfrak{S}}}_{n}$ of elements $\sigma$ such that $\sum_{i=1}^{n}(\sigma(i)-i)=0$. It is immediate to check that $f$ extends to a morphism of groups $W_{n}\rtimes\langle c\rangle\to\hat{{\mathfrak{S}}}_{n}$.

Consider $\sigma\in\hat{{\mathfrak{S}}}_{n}$ and let $N=\sum_{i=1}^{n}(\sigma(i)-i)$. Note that $n|N$. Put $\sigma^{\prime}=\sigma f(c)^{-N/n}$. We have $\sigma^{\prime}\in f(W_{n})$, so $f$ is surjective. Let $\sigma=f(wc^{d})$. We have $\sum_{i=1}^{n}(\sigma(i)-i)=nd$. So, if $\sigma=1$, then $d=0$, hence $w\in\ker(f)\cap W_{n}=1$. This shows that $f$ is injective. ∎