\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