Past exam of the mathematics course of the University of Cambridge / 2002 / ia / Paper 1 / 1B / a / Solution

For a finite group $G$ acting on a set $X$, write $G_x=\{g\in G:gx=x\}$ for the stabilizer subgroup and $Gx=\{gx:g\in G\}$ for the orbit of $x$. The orbit-stabilizer theorem givesIndeed, $g{G}_x\mapsto gx$ is a bijection from the left cosets of $G_x$ onto $Gx$: two images agree precisely when $h^{-1}g\in G_x$. Each coset has $|G_x|$ elements, proving the formula.