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:
is a bijection, and hence
is a class function ,
i.e. it is constant
on the conjugacy classes of 
.
Lemma
The mapping
. 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.