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 $x$.

The twelve rotational symmetries of a tetrahedron act on its six edges. Their edge cycle patterns are six fixed edges for the identity, two three-cycles for each of eight rotational symmetries of order three, and two fixed edges plus two transpositions for each of three half-turns. Burnside's lemma therefore counts edge colourings with $q$ labelled colours by $(q^6+8q^2+3q^4)/12$.

Pair a tetrahedron vertex with its opposite face. This is an equivariant correspondence under all tetrahedral symmetries, so three face colors and two vertex colors become six combined symbols at each of four positions. Burnside's lemma on the alternating group $A_4$ gives $(6^4+3\cdot 6^2+8\cdot 6^2)/12=141$. Including reflections gives $S_4$ and reduces the count to multisets of four symbols from six, namely $\left(\genfrac{}{}{0ex}{}{9}{4}\right)=126$.

Suppose a finite group acts on two finite position sets $A$ and $B$. Color the first set using $r$ colors and the second using $s$ colors, with both colorings moved by the same group element. If $c_A(g)$ and $c_B(g)$ are the cycle counts of its two induced permutations, the number of simultaneous colorings fixed by $g$ is $r^{c_A(g)}{s}^{c_B(g)}$: each cycle must be monochromatic, and different cycles can be colored independently. Burnside's lemma therefore givesfor the number of orbits. When the two position actions are equivariantly identified, this simplifies to the usual coloring count with $rs$ combined colors. For example, a tetrahedron's face is paired with its opposite vertex, and this pairing commutes with every symmetry.

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.