# When two singularities $\mathbb C^n/G$ and $\mathbb C^n/G'$ are the same?

Let us consider two singularities $\mathbb C^n/G$ and $\mathbb C^n/G'$, where $G$ and $G'$ are finite subgroups of $\mathrm{GL}(n,\mathbb{C})$ acting linearly.

It is easy too see, that a different groups may give the same singularities. For example, $G'=G/\langle g\rangle$, where $\langle g\rangle$ is a subgroup of $G$ generated by a reflection $g$.

Which groups $G$ and $G'$ give the same singularity?

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## 1 Answer

The answer is provided by the following

Prill's isomorphism criterion. Let $G_1, G_2 \subset \textrm{GL}(n, \mathbb{C})$ be two finite subgroups, $n \geq 2$. Assume moreover that they are small, i.e. without pseudo-reflections. Then the two germs of quotient singularities $$(\mathbb{C}^n/G_1, 0), \quad (\mathbb{C}^n/G_2, 0)$$ are analitically isomorphic if and only if $G_1$ and $G_2$ are conjugated in $\textrm{GL}(n, \mathbb{C})$.

The reference is

D. Prill: Local classification of quotients of complex manifolds by discontinuous groups, Duke Mathematical Journal 34 Number 2 (1967), 375-386.

See also this paper, Theorem 1.2 page 126.

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Furthermore, Prill proves that any $\mathbb{C}^n/G$ is isomorphic to $\mathbb{C}^m/H$ for some small subgroup $H$ of $GL(m,\mathbb{C})$. – Graham Leuschke Nov 22 '11 at 0:59