Burnside lemma

For a finite group $G$ acting on a finite set $X$, the number of orbits isIt follows by counting pairs $(g,x)$ with $gx=x$ first by $g$ and then by $x$.

If a finite group acts two-transitively on $X$, thenwhere $V$ is irreducible and nontrivial. The character norm is the number of orbits on $X\times X$, namely the diagonal and its complement.

The action of $G{L}_2({\mathbb{F}}_q)$ on ${\mathbb{P}}^1({\mathbb{F}}_q)$ has permutation representation $1\oplus \mathrm{St}$. The Steinberg constituent is irreducible of dimension $q$ and has character values $q,0,1,-1$ on scalar, nontrivial Jordan, split regular semisimple, and nonsplit semisimple elements respectively.