Edits to group theory handout

Advanced/Group Theory/parts/02 isomorphism.tex

@@ -1,7 +1,12 @@
-\section{Isomorphism}
+\section{Isomorphisms}
 
 \definition{}
-We say two groups are \textit{isomorphic} if we can create a bijective mapping between them.
+We say two groups are \textit{isomorphic} if we can create a bijective mapping between them while preserving multiplication structure. This mapping is called an \textit{isomorphism}.\\
+
+\vspace{2mm}
+
+This means that if groups $A$ and $B$ are isomorphic under $f$, \\
+$a_1 \ast a_2 = a_3$ in A implies that $f(a_1) \ast f(a_2) = f(a_3)$ in B.
 
 \problem{}
 Recall your tables from \ref{modtables}: \\
@@ -18,39 +23,46 @@ Recall your tables from \ref{modtables}: \\
 \begin{tabular}{c | c c c c}
 	\times & 1 & 2 & 3 & 4 \\
 	\hline
-	1 & 1 & 2 & 4 & 3 \\
-	2 & 2 & 4 & 3 & 1 \\
-	3 & 4 & 3 & 1 & 2 \\
-	4 & 3 & 1 & 2 & 4 \\
+	1 & 1 & 2 & 3 & 4 \\
+	2 & 2 & 4 & 1 & 3 \\
+	3 & 3 & 1 & 4 & 2 \\
+	4 & 4 & 3 & 2 & 1 \\
 \end{tabular}
 \end{center}
-Are $(\mathbb{Z}/4, +)$ and $( (\mathbb{Z}/5)^\times, \times)$ isomorphic? If they are, find a bijection that maps one to the other.
+Are $(\mathbb{Z}_4, +)$ and $(\mathbb{Z}_5^\times, \times)$ isomorphic? If they are, find a bijection that maps one to the other.
 \vfill
 
 \problem{}
-Let groups $A$ and $B$ be isomorphic, and let $f: A \to B$ be a bijection. Show that $f(e_A) = e_B$, where $e_A$ and $e_B$ are the identities of $A$ and $B$.
+Let groups $A$ and $B$ be isomorphic under $f$. Show that $f(e_A) = e_B$, where $e_A$ and $e_B$ are the identities of $A$ and $B$.
 \vfill
 
 \problem{}
-Let groups $A$ and $B$ be isomorphic, and let $f: A \to B$ be a bijection. \\
+Let groups $A$ and $B$ be isomorphic under $f$. \\
 Show that $f(a^{-1}) = f(a)^{-1}$ for all $a \in A$.
 \vfill
 
 \problem{}
-Let groups $A$ and $B$ be isomorphic, and let $f: A \to B$. Show that $f(a)$ and $a$ have the same order.
+Let groups $A$ and $B$ be isomorphic under $f$. Show that $f(a)$ and $a$ have the same order.
 
 \vfill
 \pagebreak
 
-\problem{}
+\problem{}<howmanygroups>
 Find all distinct groups of two elements. \\
 Find all distinct groups of three elements. \\
 Groups that are isomorphic are not distinct.
 
+\begin{solution}
+	There is only one nonisomorphic two-element group. \\
+	The same is true of a three-element group. \\
+
+	See \texttt{https://oeis.org/A000001}, titled \say{Number of groups of order n}
+\end{solution}
+
 \vfill
 
 \problem{}
-Show that the groups $(\mathbb{R}, +)$ and $(\mathbb{Z}^+, \times)$ are isomorphic.
+Show that the groups $(\mathbb{R}, +)$ and $(\mathbb{R}^+, \times)$ are isomorphic.
 \vfill
 
 \pagebreak