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

X_g' -> X_gg'g^-1 :x -> gx

is a bijection, and hence

c:G -> N :g -> | X_g |

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 | X_g' | = | X_gg'g^-1 | .

Proof: That x -> gx establishes a bijection between X_g' and X_gg'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.

The permutation character