The following is a very important application of actions on 
-subsets:
The regular representation of 
yields, in accordance with 
, the -sets 
,
for 
. If is finite and 
a prime
dividing 
, say 
, where does not divide 
, then we can put 
and
consider the particular -set 
,
as H. Wielandt did in his famous
proof of Sylow's Theorem
in order to show that possesses
subgroups of order 
. His argument runs as follows:
is the exact power of dividing the order of
. This is
clear from
as each power of contained in the denominator cancels.
Thus -subsets 
exist, the orbit length
of which is not divisible by 
.
We consider such an and show that its stabilizer 
is of
order by proving that is both an upper and lower bound:
For each 
and 
we have that 
, hence
On the other hand, the fact that does not divide the orbit length
yields
This proves the first item of
.
Sylow's Theorem
Assume to be a finite group
and to be a prime divisor of its order. Then
The subgroups contains
subgroups of order , for each power dividing its
order .
of the maximal -power order are called
the Sylow -subgroups
of 
They have the following properties:
-subgroup 
of is contained
in a suitable Sylow -subgroup 
.
-subgroups of are conjugate subgroups.
The proof of the second and third item follows from a consideration of double
cosets. Assume a -subgroup of and a Sylow -subgroup 
Then
we derive from that
where 
denotes a transversal of 
If all the intersections in the denominator on the right hand side were proper subgroups
of 
then the right hand side were divisible by 
which contradicts the left hand
side. Hence there must exist a 
such that 
Since
is a Sylow -subgroup, too, is contained in a Sylow subgroup, which
proves the second item.
The third item follows by taking for a Sylow -subgroup 
shows that 
for a suitable 
where 
denotes a transversal of 
This example shows clearly that the consideration of suitable group actions can be very helpful, at least in group theory. Applications to other fields of mathematics will follow soon.
Exercises
E 
.
Prove that the number of Sylow -subgroups divides the order of
and is congruent 1 modulo for each prime divisor of the order of