| | | **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 _{G}X, in short.

harald.fripertinger@kfunigraz.ac.at,

last changed: August 28, 2001

| | | **The permutation character** |