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definition:subgroup

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definition:subgroup [2013/08/20 16:11]
bmwoodruff created 		definition:subgroup [2013/08/21 13:09] (current)
bmwoodruff
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 ====== Subgroup ====== ====== Subgroup ======
 ====Definition====  ====Definition==== 
-Let $H$ be a subset of a [[definition:group]] $G$.  We say $H$ is a subgroup of $G$ if $H$ is a group itself when using the the multiplication structure of $G$ restricted to $H$. We'll write $H\leq G$ to mean $H$ is a subgroup of $G$.+Let $(G,\cdot)$ be a [[definition:group]], and let $H$ be a nonempty subset of $G$. Then $H$ is called a $\textdef{subgroup}$ of $G$ if the following hold: 
 +  - $\textbf{[Closure]}$ for all $h,k\in H$ one has $h\cdot k\in H$, and 
 +  -   $(H,\cdot)$ is a group. 
 +When $H$ is a subgroup of $G$, we write $H\le G$. Any subgroup of $G$ that is not equal to $G$ itself is called a $\textdef{proper subgroup}$. The subset of $G$ consisting of just the identity is called the $\textdef{trivial subgroup}$. 
 + 
 +====Definition 2====  
 +Let $H$ be a subset of a [[definition:group]] $G$.  We say $H$ is a subgroup of $G$ if $H$ is a group itself when using the multiplication structure of $G$ restricted to $H$. We'll write $H\leq G$ to mean $H$ is a subgroup of $G$.
  
 ---- ----
 ==== Remarks ====  ==== Remarks ==== 
-   
   * This would be a good place to add links to any subgroup theorems.    * This would be a good place to add links to any subgroup theorems. 
  
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 %%%%% %%%%%
 % DEPENDENCIES  % DEPENDENCIES 
-% RequiredMacros: \DeclareMathOperator{\syl}{Syl} +% RequiredMacros: \newcommand{\textdef}[1]{\textit{#1}} 
 %%%%% %%%%%
 \begin{definition} \begin{definition}
-Please add comments above to adjust this definition until we come upon one we agree on.+Let $(G,\cdot)$ be a group, and let $H$ be a nonempty subset of $G$. Then $H$ is called a \textdef{subgroup} of $G$ if the following hold: 
 +\begin{enumerate} 
 +\item \textbf{[Closure]} for all $h,k\in H$ one has $h\cdot k\in H$, and 
 +\item  $(H,\cdot)$ is a group. 
 +\end{enumerate} 
 +When $H$ is a subgroup of $G$, we write $H\le G$.  Any subgroup of $G$ that is not equal to $G$ itself is called a \textdef{proper subgroup}. The subset of $G$ consisting of just the identity is called the \textdef{trivial subgroup}.
 \end{definition} \end{definition}
 </file> </file>
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-{{tag>definition}}+{{tag>definition needsreview}}
definition/subgroup.1377029506.txt.gz · Last modified: 2013/08/20 16:11 by bmwoodruff

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