“Divide by the number of symmetries” is one of the most tempting shortcuts in combinatorics. It works only when every object has an orbit of the same size.

The warning sign

If some configurations look more symmetric than others, their stabilizers are different. A generic necklace may have $n$ distinct rotations, while a repeating necklace has fewer. Dividing the raw count by $n$ treats those cases as if their orbits were equal.

Burnside's lemma replaces that assumption with an average:

$$|X/G|=\frac{1}{|G|}\sum _{g\in G}|\mathrm{Fix}(g)|.$$

Rather than counting orbits directly, count how many configurations each symmetry leaves unchanged.

Why the average is exact

Count pairs $(g,x)$ for which $g$ fixes $x$ in two ways. Summing by group elements gives the numerator above. Summing by objects gives the total size of all stabilizers. The orbit–stabilizer theorem then contributes exactly $|G|$ for each orbit.

This is a useful recurring pattern: when direct division fails, search for a double count whose local irregularities average out.