handouts/Advanced/Group Theory/parts/02 isomorphism.tex

\section{Isomorphisms}

\definition{}
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}: \\
\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 & 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.
\vfill

\problem{}
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 under $f$. \\
Show that $f(a^{-1}) = f(a)^{-1}$ for all $a \in A$.
\vfill

\problem{}
Let groups $A$ and $B$ be isomorphic under $f$. Show that $f(a)$ and $a$ have the same order.

\vfill
\pagebreak

\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 $(\mathbb{R}^+, \times)$ are isomorphic.
\vfill

\pagebreak