Sylow p-subgroup - Knowlpedia

Let $G$ be a finite group and let $p$ be a prime. Write

|G|=p^n m

with $n\ge 0$ and $p\nmid m$ (so $p^n$ is the largest power of $p$ dividing $|G|$ ). A Sylow $p$ -subgroup of $G$ is a subgroup $P\le G$ such that $|P|=p^n$ . In particular, every Sylow $p$ -subgroup is a $p$ -group.

Examples

In $S_3$ (order $6=2\cdot 3$ ), Sylow $2$ -subgroups have order $2$ (each is generated by a transposition), and the unique Sylow $3$ -subgroup is $A_3$ .
In $A_4$ (order $12=2^2\cdot 3$ ), a Sylow $3$ -subgroup has order $3$ and is generated by a $3$ -cycle.
If $|G|=p^n$ , then $G$ itself is the unique Sylow $p$ -subgroup.

Remarks

The first Sylow theorem guarantees that Sylow $p$ -subgroups exist whenever $p$ divides $|G|$ . The second Sylow theorem states that any two Sylow $p$ -subgroups are conjugate (in the sense of conjugation of subgroups).