= Rank of a permutation action
{title2=$\operatorname{rank}(G\curvearrowright T)=\langle\chi,\chi\rangle$}
The rank is the number of orbits on ordered pairs under the diagonal <group action>. Its fixed-pair count at $g$ is $\chi(g)^2$, since the <permutation character> counts fixed points. <Burnside's lemma> identifies its average with the orbit count, while real-valuedness makes that average the <character inner product> of $\chi$ with itself. For a transitive action on at least two points, rank two is equivalent to transitivity on distinct ordered pairs.
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