The Cauchy-Frobenius Lemma 2Actions of groupsThe Cauchy-Frobenius LemmaThe 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@kfunigraz.ac.at,
last changed: August 28, 2001

The Cauchy-Frobenius Lemma 2Actions of groupsThe Cauchy-Frobenius LemmaThe permutation character