is injective if and only if the mapping described
in the first item of 
is
injective and corresponding cyclic factors of 
and
have the same length. The number of such mappings is
while the second item of
says that we have to multiply this number by 
in order to get the number 
of fixed points of 
on 
. Thus we have proved
and hence, by
restriction, the numbers of fixed points of
.
Corollary The number of fixed points of on
is
and of 
are:
An application of the Cauchy-Frobenius Lemma yields the desired number of injective symmetry classes:
so that we obtain by restriction the number of injective
and the number of injective
.
Theorem The number of injective
-classes is
-classes
-classes
Try to compute the number of injective symmetry classes for various group actions.
Exercises
E 
.
is called 
- fold transitive
if and only if
the corresponding action of on the set of 
of injective mapping is
transitive:
Prove that, in case of a transitive action , this is equivalent to:
E 
.
Prove that 
is divisible by 
if is finite
and -fold transitive.
E 
.
Show that 
is 
-fold transitive on
while 
is 
-fold transitive on , but not 
-fold
transitive, for 