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I haven't found satisfying answer yet. Beside those typical examples like how many colourings of a cube is there up to rotations or how many graphs are there on n vertices up to isomorphism etc., is there any "real" usage of Burnside's lemma in advanced math? All examples I could find seem to me more like funny riddles you can impress your friends with but they are not quite useful for anything.

Tusau
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  • See: https://math.mit.edu/~apost/courses/18.204_2018/Jenny_Jin_paper.pdf – MandelBroccoli Jun 03 '23 at 01:36
  • Here is a result from "probabilistic graph theory." We say that a graph is "asymmetric" if the automorphism group of the graph is trivial. The probability that a randomly-chosen, connected graph with $n$ vertices is asymmetric approaches zero as $n$ approaches infinity. This result uses Burnside's Lemma, but the proof is a bit long for me to write down as an answer. – Chris Jun 03 '23 at 04:27
  • This argument uses the lemma to prove that, if $p\nmid q-1$, then $C_p$ acts trivially on $C_q$, without knowing anything about $\operatorname{Aut}(C_q)$ (not even its order, let alone its isomorphism class): https://math.stackexchange.com/a/4609764/1092170 – citadel Jun 03 '23 at 06:04

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