# Sylow subgroup

From Citizendium

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 , with *t* coprime to *p*

- Theorem 1. There exists at least one subgroup of
*G*of order , 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.

## References

- M. Aschbacher (2000).
*Finite Group Theory*, 2nd ed, 19. ISBN 0-521-78675-4.