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====== Sylow's theorem ======
$\DeclareMathOperator{\syl}{Syl}$
**Theorem.** Let $G$ be a finite [[Definition:Group|group]] and $p$ a prime. Write $|G| = np^k$ with $(n,p) = 1$. Then
  - $G$ acts [[Definition:Transitive Group Action|transitively]] on $\syl_p(G)$ by [[Definition:Conjugation|conjugation]] with $|\syl_p(G)| \equiv 1$ modulo $p$, and
  - every $P \in \syl_p(G)$ has [[Definition:Order of a Group|order]] $p^k$.

----
==== Remarks ==== 
  * $(n,p)$ denotes the greatest common divisor of $n$ and $p$. 
  * $\syl_p(G)$ denotes the collection of [[Definition:Sylow p-Subgroup|Sylow $p$-subgroups]] of $G$.

----
==== $\LaTeX$ version ====
<code>
%%%%%%%%%%
% DEPENDENCIES
% RequiredMacros: \DeclareMathOperator{\syl}{Syl} 
%%%%%%%%%%
\begin{theorem}[Sylow's thereom]
Let $G$ be a finite group and $p$ a prime. Write $|G| = np^k$ with $(n,p) = 1$. Then
\begin{enumerate}
\item $G$ acts transitively on $\syl_p(G)$ by conjugation with $|\syl_p(G)| \equiv 1$ modulo $p$, and
\item every $P \in \syl_p(G)$ has order $p^k$.
\end{enumerate}
\end{theorem}
</code>

----
==== External links ====
  * [[wp>Sylow_theorems]]

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