we use the preceeding corollaries
together with an application of the Principle of Inclusion and
Exclusion in order to get rid of the nonsurjective fixed points. We denote
by 
the set of points 
contained in the 
-th cyclic
factor of 
and put, for each index set 
:
Then, by the Principle of Inclusion and Exclusion, we obtain for the
desired number of surjective fixed points of the following expression:
Now we recall that
This set can be identified with 
, where 
denotes the product of the cyclic factors of the numbers of
which lie in 
, and where 
is the set of points contained in
these cyclic factors. Thus
We can make this more explicit by an application of 
which yields:
Putting these things together we conclude
where the middle sum is taken over all
the sequences
and
where the sum is taken over all the sequences 
.
Corollary The number of surjective fixed points of is
of
natural numbers 
such that 
(they correspond to all possible choices of out of 
, where
of the chosen cyclic factors of are 
-cycles).
Hence the numbers of surjective fixed points of 
and of amount to:
,
and 
.
An application of the Cauchy-Frobenius Lemma finally yields the desired numbers of surjective symmetry classes:
where the inner sum is taken over the sequences
and
where the last sum is to be
taken over all the sequences 
.
Theorem The number
of surjective 
-classes is
described in the corollary above. This implies, by
restriction, the equations
such that and .
Try to compute the number of surjective symmetry classes for various group actions.
Exercises
.
.