From cc42a42b013adee7520f3d83c625920dd0bd3bf5 Mon Sep 17 00:00:00 2001
From: Mark <mark@betalupi.com>
Date: Thu, 4 Jan 2024 11:34:55 -0800
Subject: [PATCH] Edits

---
 Advanced/Symmetric Group/parts/3 subgroup.tex | 50 ++++++++++++++++---
 1 file changed, 44 insertions(+), 6 deletions(-)

diff --git a/Advanced/Symmetric Group/parts/3 subgroup.tex b/Advanced/Symmetric Group/parts/3 subgroup.tex
index f694f0d..6c9d8d2 100644
--- a/Advanced/Symmetric Group/parts/3 subgroup.tex	
+++ b/Advanced/Symmetric Group/parts/3 subgroup.tex	
@@ -72,13 +72,51 @@ Show that $S_3$ is a subgroup of $S_4$.
 \pagebreak
 
 
-\problem{}<firstindex>
-How many subgroups of $S_4$ are behave like to $S_3$? \par
-\note{
-	Of course, \say{behaves like} is a very hand-wavy relationship. \\
-	Formally, this is called \textit{isomorphism}, but we'll formally define that
-	in a later lesson.
+
+\definition{}
+Let $G$ and $H$ be groups. We say that $G$ and $H$ are \textit{isomorphic} (and write $A \simeq B$) \par
+if there is a bijection $f: G \to H$ with the following properties:
+\begin{itemize}
+	\item $f(e_G) = e_H$, where $e_G$ is the identity in $G$
+	\item $f(x^{-1}) = f(x)^{-1}$ for all $x$ in $G$
+	\item $f(xy) = f(x)f(y)$ for all $x, y$ in $G$
+\end{itemize}
+
+Intuitively, you can think of isomorphism as a form of equivalence. \par
+If two groups are isomorphic, they only differ by the names of their elements. \par
+The function $f$ above tells us how to map one set of labels to the other.
+
+
+
+\problem{}
+Show that $\mathbb{Z}_7^\times$ and $\mathbb{Z}_9^\times$ are isomorphic.
+\hint{
+	Build a bijection with the above properties. \\
+	Remember that a group is fully defined by its multiplication table.
 }
+\vfill
+
+
+\problem{}
+Show that $\mathbb{Z}_{10}^\times$ and $\mathbb{Z}_4^\times$, and $\mathbb{Z}_3$ are isomorphic.
+\hint{
+	Build a bijection with the above properties. \\
+	Remember that a group is fully defined by its multiplication table.
+}
+
+
+\vfill
+
+\problem{}
+Show that isomorphism is transitive. \par
+That is, if $A \simeq B$ and $B \simeq C$, then $A \simeq C$.
+
+\vfill
+\pagebreak
+
+
+\problem{}<firstindex>
+How many subgroups of $S_4$ are isomorphic to $S_3$? \par
 
 \vfill