.
Corollary
Let 
be a finite action and let 
denote a
transversal of the conjugacy classes of G. Then
Here is the faster version of the Cauchy-Frobenius Lemma.
Another formulation of the Cauchy-Frobenius Lemma makes use of the
permutation representation 
defined by the action
in question. (Actually in all our programs we apply this version of the Lemma.)
The permutation group 
which is the image of 
under this representation, yields the action 
of on 
,
which has the
same orbits, and so
we also have:
where
.
Corollary
If denotes a finite -set, then (for
any group ) the following identity holds:
denotes a transversal of the conjugacy classes of .
Exercises
E 
.
Let be finite and transitive. Consider an
arbitrary 
and prove that
