Sylow p-subgroup

Definition

Let $G$ be a finite group and let $p$ be a prime number. Write $|G| = p^n m$ with $n \geq 0$ and $p \nmid m$. A Sylow $p$-subgroup of $G$ is a subgroup of $G$ of order $p^n$. The set of the Sylow $p$-subgroups of $G$ is written $\operatorname{Syl}_p(G)$, and its size is written $n_p$.

Remarks

A Sylow $p$-subgroup is a maximal p-subgroup of $G$. If $p$ does not divide $|G|$, then the only Sylow $p$-subgroup is $\{e\}$. In $S_3$, the Sylow $3$-subgroup is $A_3$, and there are three Sylow $2$-subgroups, each of order $2$.

Explanations

L1 human

A Sylow subgroup takes all the factors $p$ of the order of the group at once. It is as large as a $p$-subgroup can be by Lagrange's theorem. The Sylow theorems say that such subgroups exist and that they are all alike.