The Sylow theorems are the most powerful general tools for understanding finite groups. They guarantee the existence of subgroups of prime-power order and constrain how many there can be.
Definition 11.3.1 (Sylow -subgroup). Let be a finite group and a prime dividing . Write where . A Sylow -subgroup of is a subgroup of order .
Note. The converse to Lagrange’s theorem fails in general: if , there need not exist a subgroup of order . For instance, has no subgroup of order 6, and has no subgroup of order 30. The Sylow theorems say this failure does not occur for prime-power divisors.
Theorem 11.3.1 (Sylow Theorems (Sylow, 1872)). Let be a finite group with ,.
For every prime dividing , a Sylow -subgroup exists.
All Sylow -subgroups are conjugate: if are both Sylow -subgroups, then for some . Moreover, any subgroup with a power of is contained in some Sylow -subgroup.
Let be the number of Sylow -subgroups. Then and (equivalently, ).
Example 11.3.1 (Groups of order 15). Let . By Sylow (3): and , so . Similarly, and , so . Both Sylow subgroups are normal (unique Sylow subgroups are always normal, since conjugation permutes Sylow subgroups). Let be the Sylow 3-subgroup and the Sylow 5-subgroup. Then (orders are coprime) and . So . Every group of order 15 is cyclic.