Burnside's Lemma and Geometry - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T03:59:22Z http://mathoverflow.net/feeds/question/39415 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/39415/burnsides-lemma-and-geometry Burnside's Lemma and Geometry Michele Triestino 2010-09-20T18:32:26Z 2010-09-20T19:07:28Z <p>I think one of the most interesting results in Elementary Group Theory is the so-called "<a href="http://en.wikipedia.org/wiki/Burnside%27s_lemma" rel="nofollow">Burnside's Lemma</a>", counting the numbers of orbits of a (finite) group action.</p> <p>I wonder if there is any (interesting) application in Elementary Geometry (I mean Euclidean, hyperbolic or elliptic geometry).</p> <p>Searching on Google, I've found the article "<a href="http://users.wpi.edu/~bservat/strippat.pdf" rel="nofollow">Applying Burnside’s lemma to a one-dimensional Escher problem</a>" by T. Pisanski, but it sounds to me rather a combinatorial result.</p> http://mathoverflow.net/questions/39415/burnsides-lemma-and-geometry/39421#39421 Answer by Benoît Kloeckner for Burnside's Lemma and Geometry Benoît Kloeckner 2010-09-20T19:07:28Z 2010-09-20T19:07:28Z <p>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.</p>