, 
, 
and 
on 
, obtained from given actions 
and 
. The
orbits of these groups are called symmetry classes of mappings.
If
we want to be more explicit, we call them -classes,
-classes, -classes and -classes, respectively. Their total number can be
obtained by an application of the Cauchy-Frobenius Lemma as soon as we know
the number of fixed points for each element of the respective group. In order
to derive these numbers we characterize the fixed points of each 
on and then we use the natural embedding of
, and
in as described above.
Thus the following
lemma will turn out to be crucial:
is the disjoint cycle decomposition of
.
Lemma
Consider an 
, an element
of and assume that
, the permutation of 
which corresponds to 
.
Then 
is a fixed point of if and only if
the following two conditions
hold:
is a fixed point of the
cycleproduct 
:
arise from the
values according to
the following equations:
Proof: 
says that is fixed under if and only if
its values 
satisfy the equations
where 
denotes the length of the cyclic factor of
containing the point 
. Hence in particular the following must be
true:
which means that is a fixed point of , as claimed.
Thus any fixed clearly has the stated properties, and vice
versa.
This, together with the Cauchy-Frobenius Lemma, yields the number of
-classes on , and the restrictions to the subgroups ,
and
give the numbers of -, - and -classes
on :
The restriction to
and
.
Theorem
If both and
are finite actions, then we
obtain the following expression
for the total number of orbits of the corresponding action of
on :
, and according to yields:
In order to apply these results to a
specific case it remains to evaluate
and 
or 
which still can be quite cumbersome as the following example
shows.
Try to
compute the number of symmetry classes of mappings
for various group actions. Furthermore you can compute a
transversal of G-classes
on .