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Here is a teeny tiny toy version of the Lefschetz fixed point theorem: let $f : S \to S$ be an endomorphism of a finite set and let $K[f] : K[S] \to K[S]$ be the induced linear map on free vector spaces. Then $\text{tr}(K[f])$ is the number of fixed points of $f$. This is one way to prove Burnside's lemma.