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Wednesday, December 12, 2007

Burnside's Lemma

Burnside's Lemma as it can be found on MathWorld:
Let J be a finite group and the image R(J) be a representation which is a homomorphism of J into a permutation group S(X), where S(X) is the group of all permutations of a set X. Define the orbits of R(J) as the equivalence classes under x~y ,which is true if there is some permutation p in R(J) such that p(x)=y. Define the fixed points of p as the elements x of X for which p(x)=x. Then the arithmetic mean number of fixed points of permutations in R(J) is equal to the number of orbits of R(J).

Today I deepened my understanding of Burnside's Lemma considerably.

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(Raumpatrouille – Die phantastischen Abenteuer des Raumschiffes Orion, colloquially aka Raumpatrouille Orion was the first German science fiction television series. Its seven episodes were broadcast by ARD beginning September 17, 1966. The series has since acquired cult status in Germany. Broadcast six years before Star Trek first aired in West Germany (in 1972), it became a huge success.)