Added group theory parts

Advanced/Group Theory/parts/02 isomorphism.tex

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+\section{Isomorphism}
+
+\definition{}
+We say two groups are \textit{isomorphic} if we can create a bijective mapping between them.
+
+\problem{}
+Recall your tables from \ref{modtables}: \\
+\begin{center}
+\begin{tabular}{c | c c c c}
+	+ & 0 & 1 & 2 & 3 \\
+	\hline
+	0 & 0 & 1 & 2 & 3 \\
+	1 & 1 & 2 & 3 & 0 \\
+	2 & 2 & 3 & 0 & 1 \\
+	3 & 3 & 0 & 1 & 2 \\
+\end{tabular}
+\hspace{1cm}
+\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 \\
+\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.
+\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 $B$.
+\vfill
+
+\problem{}
+Let groups $A$ and $B$ be isomorphic, and let $f: A \to B$ be a bijection. \\
+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.
+
+\vfill
+\pagebreak
+
+\problem{}
+Find all distinct groups of two elements. \\
+Find all distinct groups of three elements. \\
+Groups that are isomorphic are not distinct.
+
+\vfill
+
+\problem{}
+Show that the groups $(\mathbb{R}, +)$ and $(\mathbb{Z}^+, \times)$ are isomorphic.
+\vfill
+
+\pagebreak