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--- definition:subgroup [2013/08/20 16:12]
bmwoodruff
+++ definition:subgroup [2013/08/21 13:09] (current)
bmwoodruff
@@ Line 1: / Line 1: @@
 ====== Subgroup ======
 ====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 $G$.
 ----
@@ Line 20: / Line 26: @@
 %%%%%
 % DEPENDENCIES
-% RequiredMacros: \DeclareMathOperator{\syl}{Syl}
+% RequiredMacros: \newcommand{\textdef}[1]{\textit{#1}}
 %%%%%
 \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}
 </file>
@@ Line 32: / Line 43: @@
-{{tag>definition}}
+{{tag>definition needsreview}}

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