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theorem:sylow_s_theorem

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theorem:sylow_s_theorem [2013/08/08 07:33]
joshuawiscons 		theorem:sylow_s_theorem [2013/08/13 11:29] (current)
joshuawiscons
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-====== Sylow's Theorem ====== +====== Sylow's theorem ====== 
-==== Statement ==== +$\DeclareMathOperator{\syl}{Syl}$ 
-Let $G$ be a finite [[Definition:Group|group]] and $p$ a prime. Write $|G| = np^k$ with $(n,p) = 1$. Then +**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 $\textrm{Syl}_p(G)$ by [[Definition:Conjugation|conjugation]] with $|\textrm{Syl}_p(G)| \equiv 1$ modulo $p$, and +  - $G$ acts [[Definition:Transitive Group Action|transitively]] on $\syl_p(G)$ by [[Definition:Conjugation|conjugation]] with $|\syl_p(G)| \equiv 1$ modulo $p$, and 
-  - every $P \in \textrm{Syl}_p(G)$ has [[Definition:Order of a Group|order]] $p^k$.+  - every $P \in \syl_p(G)$ has [[Definition:Order of a Group|order]] $p^k$.
  
-== Notation == +---- 
-  * $\textrm{Syl}_p(G)$ denotes the collection of [[Definition:Sylow p-Subgroup|Sylow $p$-subgroups]] of $G$.+==== 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 ==+---- 
 +==== $\LaTeX$ version ====
 <code> <code>
 %%%%%%%%%% %%%%%%%%%%
 % DEPENDENCIES % DEPENDENCIES
-% --RequiredMacros: \DeclareMathOperator{\syl}{Syl} +% 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 Let $G$ be a finite group and $p$ a prime. Write $|G| = np^k$ with $(n,p) = 1$. Then
 \begin{enumerate} \begin{enumerate}
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 \item every $P \in \syl_p(G)$ has order $p^k$. \item every $P \in \syl_p(G)$ has order $p^k$.
 \end{enumerate} \end{enumerate}
 +\end{theorem}
 </code> </code>
  
-==== External Links ==== +---- 
-  * [[http://en.wikipedia.org/wiki/Sylow_theorems|Sylow's Theorem on Wikipedia]]+==== External links ==== 
 +  * [[wp>Sylow_theorems]]
theorem/sylow_s_theorem.1375961627.txt.gz · Last modified: 2013/08/08 07:33 by joshuawiscons

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