Sylow subgroup

In group theory, a Sylow subgroup of a group is a subgroup which has order which is a power of a prime number, and which is not strictly contained in any other subgroup with the same property. Such a subgroup may also be called a Sylow p -subgroup or a p -Sylow subgroup.

The Sylow theorems describe the structure of the Sylow subgroups. Suppose that p is a prime which divides the order n of a finite group G, so that $n=p^{s}t$, with t coprime to p

• Theorem 1. There exists at least one subgroup of G of order $p^{s}$, which is thus a Sylow p-subgroup.
• Theorem 2. The Sylow p-subgroups are conjugate.
• Theorem 3. The number of Sylow p-subgroups is congruent to 1 modulo p.

The first Theorem may be regarded as a partial converse to Lagrange's Theorem.