The permutation character



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The permutation character

The next remark helps considerably to shorten the calculations necessary for applications of this lemma. It shows that we can replace the summation over all by a summation over a transversal of the conjugacy classes, as the number of fixed points turns out to be constant on each such class:

. Lemma   The mapping

is a bijection, and hence

is a class function  , i.e. it is constant on the conjugacy classes of . More formally, for any , we have that .

Proof: That establishes a bijection between and is clear from the following equivalence:

The mapping is called the character of the action of on , or of , in short.



Herr Fripertinger
Sun Feb 05 18:28:26 MET 1995