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 $g$.



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> Which groups $G$ and $G'$ give the same singularity?
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When two singularities $\mathbb C^n/G$ and $\mathbb C^n/G'$ are the same?
The answer is provided by the following



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> **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 $\textrm{GL}(n, \mathbb{C})$.
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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](http://books.google.com/books?id=PqKXHI4PQd0C&pg=PA126&lpg=PA126&dq=quotient+singularities+are+isomorphic+groups+are+conjugate&source=bl&ots=IQzWVa2wXG&sig=npdkiL0TxmNRCY8IrhYrnP3D34g&hl=it&ei=r6rKTp_aF8WmhAeqsNjQDw&sa=X&oi=book_result&ct=result&resnum=1&ved=0CB8Q6AEwAA#v=onepage&q=quotient%2520singularities%2520are%2520isomorphic%2520groups%2520are%2520conjugate&f=false), Theorem 1.2 page 126.


