56 lines
1.4 KiB
TeX
56 lines
1.4 KiB
TeX
\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 |