First Sylow Theorem

Let be a Group.
The set of Sylow Subgroups is nonempty.

Proof

Consider , the set of all subsets of of size .
Note that which is not divisible by .
Consider the natural (left multiplication) Group Action of on .
It has at least one Orbit not divisible by .
Suppose that is the orbit of .
Now by Orbit-Stabilizer Theorem
so
so .
But also
because .
So
so .