Definition. Let $G$ be a group and $p$ a prime. A $p$-subgroup $P \le G$ is called a Sylow $p$-subgroup of $G$ if $P$ is not properly contained in any other $p$-subgroup of $G$, i.e. $P$ is a maximal $p$-subgroup of $G$.

• By Zorn's lemma, Sylow $p$-subgroups exist in any group; however, they may be trivial or equal to the whole group $G$.
• The above definition naturally extends to the idea of a Sylow $\pi$-group where $\pi$ is any collection of primes. See the remarks following the definition of a $p$-group.

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\begin{definition}
Let $G$ be a group and $p$ a prime. A $p$-subgroup $P \le G$ is called a \textDef{Sylow $p$-subgroup} of $G$ if $P$ is not properly contained in any other $p$-subgroup of $G$, i.e. $P$ is a maximal $p$-subgroup of $G$.
\end{definition}

• Sylow theorems