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Every nonidentity biholomorphism in the finite group $G$ has only finitely many fixed points, by the identity theorem on a Riemann surface and compactness. Because the group action is a faithful group action, the union of these finite sets does not exhaust $X$. A point outside this union has trivial stabilizer subgroup, so its group orbit has $|G|$ points by the orbit-stabilizer theorem. The generic fiber of a function of the projection is such an group orbit, givingMore precisely, put $H=G_x$ and $m=|H|$. Differentiation at $x$ gives a group homomorphism $H\to {\mathbb{C}}^{\times }$. It is injective: if a nonidentity finite-order germ had the form $h(z)=z+a_k{z}^k+O(z^{k+1})$ with $k\ge 2$ and $a_k\ne 0$, then its $q$th iterate would begin $z+q{a}_k{z}^k$, so no positive iterate could be the identity. A germ equal to the identity gives the identity on $X$ by the identity theorem on a Riemann surface. Thus $H$ is a cyclic group, with derivative image the $m$th roots of unity.

For completeness, an explicit local coordinate linearizing $H$ isIt has $w^{\prime }(x)=1$, and reindexing the sum gives $w(hp)=h^{\prime }(x)w(p)$. Therefore a coordinate on the local quotient space is $w^m$. Other translates of a sufficiently small neighbourhood of $x$ are disjoint unless their translating element lies in $H$, so this local quotient space is also a chart for $X/G$. The local cyclic quotient of a Riemann surface consequently gives

Choose a local coordinate $w$ with $w(p)=0$. Differentiation at $p$ gives a homomorphismIt is injective. Indeed, a nonidentity Möbius transformation fixing $p$ and having derivative one there is parabolic, hence conjugate to a nonzero translation and of infinite order; no such element can belong to the finite group $H$. Every finite subgroup of ${\mathbb{C}}^{\times }$ is a cyclic group of roots of unity, so $H$ is cyclic. Write $n=|H|$.

The averaged coordinatehas derivative one at $p$, so it is a coordinate on a smaller neighbourhood. For $g\in H$, reindexing by $k=hg$ givesAfter shrinking to an $H$-invariant disc $U$, the group therefore acts as all rotations $z\mapsto {\zeta }^{j}z$, where $\zeta$ is a primitive $n$th root of unity. The invariant coordinateidentifies $H\backslash U$ with a disc and gives it a Riemann-surface chart. In these coordinates the quotient map is exactlyThis is the local cyclic quotient of a Riemann surface.