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I think one of the most interesting results in Elementary Group Theory is the so-called "Burnside's Lemma", counting the numbers of orbits of a (finite) group action.

I wonder if there is any (interesting) application in Elementary Geometry (I mean Euclidean, hyperbolic or elliptic geometry).

Searching on Google, I've found the article "Applying Burnside’s lemma to a one-dimensional Escher problem" by T. Pisanski, but it sounds to me rather a combinatorial result.

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  • $\begingroup$ There may be something in the book by Paul B Yale, Geometry and Symmetry, published in 1968 (2nd edition, Dover, 1988). Well, it has the word "Geometry" in the title, and Burnside's Lemma is discussed in the 1st chapter. $\endgroup$
    – Gerry Myerson
    Commented Sep 21, 2010 at 6:30
  • $\begingroup$ "Burnside's Lemma" is not due to Burnside according to a 1979 paper of P. Neumann called (appropriately) "A lemma that is not Burnside's". Neumann attributes the result to Cauchy and Frobenius. $\endgroup$
    – Marty Isaacs
    Commented Jan 31, 2012 at 19:01
  • $\begingroup$ @Marty Thank you for your comment, that's why I wrote "so-called" $\endgroup$
    – user47274
    Commented Feb 2, 2012 at 17:39

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Burnside Lemma can be used as a first step to classify all finite subgroups of $\mathrm{SO}(3)$: it gives you that there are at most $3$ orbits in the action of any finite group $G$ on the set of intersections between axes of elements of $G$ and the unit sphere.

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    $\begingroup$ Which, of course, can then be used to classify the Platonic solids. $\endgroup$
    – Qiaochu Yuan
    Commented Sep 20, 2010 at 19:38

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