Theorem. Let be a finite group and a prime. Write with . Then

1. acts transitively on by conjugation with modulo , and
2. every has order .

• denotes the greatest common divisor of and .
• denotes the collection of Sylow $p$-subgroups of .

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% DEPENDENCIES
% RequiredMacros: \DeclareMathOperator{\syl}{Syl} 
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\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}

• Sylow_theorems