Sylow's Theorems
Table of Contents
1 Definitions
Definition A p-group is a group in which the order of all of the elements in the group are powers of \( p \). For a finite group this is equivalent to \( |G| \) is a power of \( p \).
2 Sylow's Theorems
The first of Sylow's Theorems gives a partial converse of Lagrange's Theorem.
2.1 Theorem 1
Let \( G \) be a finite group, with \( |G| = p^k * m \) where \( p \not | m \), then \( \exists H \le G \) such that \( |H| = p^k \). We call \( H \) a Sylow p-subgroup.
2.2 Theorem 2
If \( H \) and \( K \) are p-subgroups of \( G \), then they are conjugate.
2.3 Theorem 3
Let \( n_p \) be the number of Sylow p-subgroups of \( G \), then the following hold:
- \( n_p | m \)
- \( n_p \equiv 1 \pmod p \)
- \( n_p = [G : N_G(P)] \) where \( P \) is any Sylow p-subgroup of \( G \).