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Published in:
1999 | OriginalPaper | Chapter
Enumeration by Stabilizer Class
Author : Adalbert Kerber
Published in: Applied Finite Group Actions
Publisher: Springer Berlin Heidelberg
Included in: Professional Book Archive
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We now consider another refinement of the Cauchy-Frobenius Lemma. It is due to Burnside, and it allows to enumerate orbits which have a given conjugacy class of subgroups as stabilizers of their elements. For example, it allows us to count the orbits of maximal length |G|, the asymmetric orbits, since these are the orbits ω with trivial stabilizers G x = 1, for each x ∈ ω. A well known formula from Galois theory turns out to be such a number of asymmetric orbits.