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--- definition:subgroup [2013/08/21 06:00]
joshuawiscons
+++ definition:subgroup [2013/08/21 13:09] (current)
bmwoodruff
@@ Line 4: / Line 4: @@
   - $\textbf{[Closure]}$ 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 $H\le G$.
+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 the multiplication structure of $G$ restricted to $H$. We'll write $H\leq G$ to mean $H$ is a subgroup of $G$.
+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 34: / Line 34: @@
 \item  $(H,\cdot)$ is a group.
 \end{enumerate}
-When $H$ is a subgroup of $G$, we write $H\le G$.
+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>

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