Let be a group and a prime. A $p$-subgroup is called a Sylow $p$-subgroup of if is not properly contained in any other $p$-subgroup of , i.e. if is a maximal -subgroup of . The collection of all Sylow -subgroups of is usually denoted .

• By Zorn's lemma, Sylow -subgroups exist in any group; however, they may be trivial or equal to the whole group .
• The above definition naturally extends to the idea of a Sylow -group where 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 \textit{Sylow $p$-subgroup} of $G$ if $P$ is not properly contained in any other $p$-subgroup of $G$, i.e. if $P$ is a maximal $p$-subgroup of $G$. The collection of all Sylow $p$-subgroups of $G$ is usually denoted $\syl_p(G)$.
\end{definition}

• Sylow theorems