Burnside's 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 $x$.

For a permutation group, $c_r(g)$ is the number of length-$r$ cycles of $g$. The cycle index records the average cycle structure. On coloring a finite underlying set, a coloring fixed by $g$ is constant on each cycle. Replacing $t_r$ by the power-sum symmetric polynomial $p_r(x)$ therefore gives the pattern inventory by the weighted Burnside lemma.

This is the sum of content monomials of colorings up to the group action. It equals the cycle index specialized at $t_r=p_r(x)$. One may first use finitely many colors; stabilization defines the corresponding homogeneous symmetric function. Calling the specialized cycle index a cycle indicator makes $Z_G=F_G$ an equality in the same variables.

For an orbit-invariant weight $w$ on a finite group action $G\curvearrowright X$, the sum of one weight per orbit is $|G{|}^{-1}{\sum }_g{\sum }_{x:gx=x}w(x)$. Double counting gives ${\sum }_x|G_x|w(x)$ before dividing by $|G|$; each orbit contributes $|G|$ copies of its common weight. Thus the usual Burnside's lemma extends to weights in any commutative coefficient ring.

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.