Past exam of the mathematics course of the University of Cambridge / 2020 / ia / Paper 2 / 6E / a

The singleton orbits are exactly $X^G$. Every other orbit represented by $x$ has size $|G|/|G_x|$ by the orbit-stabilizer theorem. Partitioning $X$ into its orbits of a group action therefore gives

For the second identity, countin two ways. Holding $g$ fixed givesHolding $x$ fixed gives $|Z|={\sum }_{x\in X}|G_x|$. Each orbit contributes $|G|/|G_x|$ points, each with stabilizer size $|G_x|$, and hence contributes $|G|$. Thus $|Z|=|G||X/G|$, proving Burnside lemma: