Because the action is free, the quotient $S_g/G$ is a closed orientable surface $S_h$, and $S_g\to S_h$ is a covering of degree $|G|$. The <Euler characteristic under a finite covering> gives
$$
2-2g=|G|(2-2h),
$$
or equivalently
$$
g-1=|G|(h-1).
$$
The quotient inherits a <hyperbolic metric>, so $h\geq2$. Hence $h-1\geq1$ and $|G|\leq g-1$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 156 / 1 / c / Solution

A hyperbolic metric is a complete Riemannian metric of constant sectional curvature $-1$. On a surface it supplies local charts modeled on the <hyperbolic plane>.

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 Incoming links: Hyperbolic metric