We now return to 
and consider its subsets consisting of the injective
and the surjective maps 
only:
It is clear
that each of these sets is both a 
-set and an 
-set and therefore it
is also an 
-set, but it will not in general
be an 
-set.
The corresponding orbits of 
and on 
are called
injective
symmetry classes, while those on 
will be called
surjective
symmetry classes. We should like to determine their number.
In order to do this we describe the fixed points
of 
on these sets to prepare an application of the
Cauchy-Frobenius Lemma. A first remark shows how the fixed points of 
on can be constructed with the aid of 
and 
,
the permutations induced by 
on 
and by 
on 
(use 
):
and
the other values of
.
Corollary If 
, then 
is fixed under if and only if
the following two conditions
are satisfied:
arise from
the values 
according to
This together with yields:
.
Corollary The fixed points of are
the which can be
obtained in the following way:
, let 
denote its length, we
associate a cyclic factor of
of length 
dividing .
is a point in this cyclic factor
of and 
a point
in the chosen cyclic factor of , then put
