Three theorems due to the Norwegian mathematician Peter Sylow which provide partial converses to Lagrange’s theorem. Let G be a group of order p^am, where p is a prime which does not divide m and a is a positive integer. Then:

• 1^ST THEOREM: G has a subgroup of order p^a. Any such subgroup is called a Sylow p-subgroup.

• 2^ND THEOREM: Any two Sylow p-subgroups are conjugate in G.

• 3^RD THEOREM: If there are n_p Sylow p-subgroups in G, then n_p divides m and p divides n_p–1.

Sylow’s theorems have more significance when G’s order is divisible by several primes, but imply nothing useful about groups whose orders are a power of a prime.