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\noindent{\Large{\textsc{Modern Algebra \hfill Assignment 8.3  \\ Spring 2023 \hfill Due March 8}}}

\large

\vspace{1cm}

\begin{exercise}
Let $G$ be a group and $H \triangleleft G$.  Use the natural epimorphism $\pi : G \to G/H$ to prove the following.
\begin{itemize}

\item[\bf a.]  If $G$ is abelian, then $G/H$ is abelian.

\item[\bf b.]  If $G$ is cyclic, then $G/H$ is cyclic.

\end{itemize}
\end{exercise}

\begin{exercise}
Let $p$ be a prime number and define
\[
R_p = \left \{ \frac{r}{s} \in \Q \, \bigg| \, p \nmid s \right \}.
\]

\begin{itemize}

\item[\bf a.]  Prove that $pR_p < R_p < \Q$.

\item[\bf b.]  Let $f: \Z \to R_p/pR_p$ be the homomorphism obtained by composing the inclusion $\Z \hookrightarrow R_p$ with the natural epimorphism $\pi : R_p \to R_p/pR_p$.  Show that $f$ is surjective.  [{\em Suggestion.}  Given $r/s \in R_p$, the condition $p \nmid s$ implies that $s \in \Z_p^{\times}$.  So we may choose $t \in \Z$ so that $st \equiv 1 \pmod{p}$.  Show that $f(rt) = \frac{r}{s} + pR_p$.]

\item[\bf c.]  Show that $\ker f = p\Z$ and use the First Isomorphism Theorem to conclude that $\Z/p\Z \cong R_p / p R_p$.

\end{itemize}

\end{exercise}

\begin{exercise}
A {\em fractional linear transformation} is a nonconstant rational function of the form $\ell(x) = \frac{ax+b}{cx+d}$ with $a,b,c,d \in \R$.  
\begin{itemize}

\item[\bf a.]  Compute $\ell'(x)$ and conclude that $\ell(x)$ is nonconstant if and only if $ad-bc \ne 0$.

\item[\bf b.]  Let $\mathcal{L}(\R)$ denote the set of fractional linear transformations.  Prove that $\mathcal{L}(\R)$ is a group under composition of functions.

\item[\bf c.]  Show that the map $f: \operatorname{GL}_2(\R) \to \mathcal{L}(\R)$ given by
\[
\left( \begin{matrix} a & b \\ c & d \end{matrix} \right) \mapsto \frac{ax +b}{cx+d}
\]
is a surjective homomorphism.

\item[\bf d.]  Show that $\ker f = Z(\operatorname{GL}_2(\R))$ and conclude that
\[
\operatorname{GL}_2(\R) / Z(\operatorname{GL}_2(\R)) \cong \mathcal{L}(\R).
\]

\end{itemize}

\end{exercise}

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