Sylow’s Theorems

Sylow’s Theorem I

Let be a group of order , where is a prime and . Then has a subgroup of order , we call it -Sylow subgroup.

Proof We use induction on . Clearly for there is nothing to prove.

Sylow’s Theorem II

Let be a finite group and a prime dividing . Then

• Any two -Sylow subgroups in a finite group are conjugate
• Any -subgroup of is contained in a -Sylow subgroup

Sylow’s Theorem III

The number of -Sylow subgroups in a finite group of order with is of the form and divides .

Proof

Corollary

A group  has exactly one -Sylow subgroup  if and only if  is normal.