Let us now consider an application of the
preceding results to the
paradigmatic 
-set 
corresponding to given 
and 
(cf. 
). This
yields a nice proof of a famous number theoretic result.
.
Example
Let 
denote
the following cyclic subgroup
of 
:
It acts on the set 
and
hence, see ,
also on the set 
, which
can be considered as the set
of all the colourings of the regular 
-gon in 
colours. For example,
in the case
when 
and 
, 
acts on the set
, consisting
of all the 32 colourings of the regular pentagon with two colours (black and
white), some of which are shown
in figure .
Figure: Three colourings of the regular 5-gon.
We now assume that is a prime. Lemma shows
that 
contains,
besides the identity element, -cycles only. The identity element
of
keeps each 
fixed, while each -cycle fixes the
monochromatic colourings only (notice that 
acts as a
clockwise rotation of the -gon after having numbered the vertices of
the -gon from 1 to , counterclockwise). Hence we obtain from the
Cauchy-Frobenius Lemma that
provided that is a prime number. This
implies that 
, and hence also 
, is congruent zero modulo
, for each positive integer . It is clear that this is then also true
for any integer 
In the case when 
is not divisible by 
we may even
divide the difference 
by 
obtaining
.
Fermat's Congruence
If is a prime number and an integer which
is prime to then
This shows that group actions can also be useful in elementary number theory, and it seems appropriate to emphasize the following immediate implication of the Cauchy-Frobenius Lemma:
.
Corollary
For any action of a finite group on a finite set 
we have the congruence
Later on we shall return to this result and we shall refine it considerably (numbers will be replaced by polynomials, and the congruence will be a congruence for the coefficients of the monomial summands). It is a very helpful tool for number theoretic purposes and shows again the efficiency of finite group actions.