Let be a finite group.
For every prime factor with multiplicity of the order of , there exists a Sylow -subgroup of , of order .
All Sylow -subgroups of are conjugate to each other, i.e.\ if and are Sylow -subgroups of , then there exists an element with .
Let be a prime such that , where . Let be the number of Sylow -subgroups of . Then:
1) , which is the index of the Sylow -subgroup in .
Theorem 3b (Proof)
We have , where is any Sylow -subgroup of and denotes the normalizer.
Let be a Sylow -subgroup of and let act on by conjugation. We have , .
By the Orbit-Stabilizer Theorem, , thus .
Let be a group which acts on a finite set . Then