Past exam of the mathematics course of the University of Cambridge / 2003 / ia / Paper 1 / 7B / i / Solution

For a group action of $G$ on a set $X$ and $x\in X$, the orbit-stabilizer theorem statesIf $G$ is finite, this becomes $\boxed{|G|=|Gx||G_x|}$. The underlying bijection sends the left coset $g{G}_x$ to $gx$. Equality of two image points is equivalent to $g_2^{-1}{g}_1\in G_x$, so this map is both well-defined and injective, and it is surjective by the definition of the group orbit.