Burnside's Lemma

Let be a Finite Group and the image be a representation which is a Homeomorphism of into aPermutation Group , where is the Group of all permutations of a Set . Define the orbits of as the equivalence classes under , which is true if there is some permutation in such that . Define the fixed points of as the elements of for which . Then the Average number of FixedPoints of permutations in is equal to the number of orbits of .

The Cauchy (1845) in obscure form and Frobenius (1887) prior to Burnside's(1900) rediscovery. It was subsequently extended and refined by Pólya (1937) for applications inCombinatorial counting problems. In this form, it is known as Pólya EnumerationTheorem.

References

Pólya, G. ``Kombinatorische Anzahlbestimmungen für Gruppen, Graphen, und chemische Verbindungen.'' Acta Math. 68, 145-254, 1937.

• Posterior Distribution
• Postnikov System
• Postulate
• Potato Paradox
• Potential Function
• Potential Theory
• Pothenot Problem
• Poulet Number
• Power
• Power Center
• Power (Circle)
• Power Curve
• Powerfree
• Powerful Number
• Power Line
• Power Point
• Power Rule
• Power Series
• Power Set
• Power Spectrum
• Power (Statistics)
• Power Tower
• Power (Triangle)
• P-Polynomial
• p(prime)-Group