   The permutation character

### 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 g ÎG 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
Xg' -> Xgg'g-1 :x -> gx
is a bijection, and hence
c:G -> N :g -> | Xg |
is a class function , i.e. it is constant on the conjugacy classes of G. More formally, for any g,g' ÎG, we have that | Xg' | = | Xgg'g-1 | .

Proof: That x -> gx establishes a bijection between Xg' and Xgg'g-1 is clear from the following equivalence:

g'x = x iff gg'g-1(gx) = gx.

The mapping c is called the character

of the action of G on X, or of GX, in short.

 harald.fripertinger "at" uni-graz.at http://www-ang.kfunigraz.ac.at/~fripert/ UNI-Graz Institut für Mathematik UNI-Bayreuth Lehrstuhl II für Mathematik
last changed: January 19, 2005   The permutation character