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\noindent
\parbox{1.95in}{\bf Math 3110}
\hfill {\Large \bf Homework \#8} \hfill
\parbox{1.95in}{\bf \hfill Due: \sout{Mon., Apr. 1${}^{\text{\bf st}}$, 2024}}\\
\force\hfill {\bf Wed., Apr. 3${}^{\text{\bf rd}}$, 2024} % some breathing room -- not totally necessary

%\vspace{0.2in}
%
%\noindent {\bf\large Name:} \quad \underline{\large\qquad This is my Name \qquad}

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\noindent
Please remember when submitting any work via email or in person to\dots\\
\force \hfill \fbox{\fbox{\bf\large\color{red}PUT YOUR NAME ON YOUR WORK!}} \hfill \force


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\begin{description}
\item[\#1] Let $H=\langle x^2 \rangle = \{1,x^2,x^4,x^6\} \subseteq D_8= \langle x,y \;|\; x^8=1, y^2=1, (xy)^2=1 \rangle = \{ 1,x,\dots,x^7,y,xy,\dots,x^7y \}$.

\begin{enumerate}[(a)]
\item Compute the (distinct) left and right cosets of $H$ to confirm this is a normal subgroup of $D_8$.
\item Write down a Cayley table for $D_8/H$.
\item Is $D_8/H$ abelian? Is it cyclic? Explain you answers.
\end{enumerate}

\item[\#2] Quotients in $\mathbb{Z}_{n}$.
   \begin{enumerate}[(a)]
   \item Let $H = \langle 18 \rangle \subseteq \mathbb{Z}_{30}$.
      \begin{enumerate}[i.]
      \item List the elements of $H$ (i.e., $H= \{ ??? \}$) and explain why it is a normal subgroup of $\mathbb{Z}_{30}$.
      \item Compute the (distinct) cosets of $H$ in $\mathbb{Z}_{30}$.
      \item Write down a Cayley table for $\mathbb{Z}_{30}/H$. What familiar group is this isomorphic to?
      \end{enumerate}
   \item Consider $k,\ell,n \in \mathbb{Z}_{>0}$ where $n=k\ell$. We attempt to define a function $\varphi:\mathbb{Z}_n \to \mathbb{Z}_{\ell}$ by $\varphi(x)=x$.\\
            {}[Using more precise notation: $\varphi(x+n\mathbb{Z})=x+\ell\mathbb{Z}$.]
            
            Prove that $\varphi$ is a {\it well-defined} homomorphism. 
            
            Determine $\varphi$'s kernel and image. Then apply the first isomorphism theorem.
   \end{enumerate}

\item[\#3] Recall $\mathrm{GL}_n(\mathbb{R})$ (equipped with matrix multiplication) is the group of $n \times n$ invertible matrices. Also, recall $\mathbb{R}_{\not=0}$ is a group under multiplication. Explain why the determinant map, $\det:\mathrm{GL}_n(\mathbb{R}) \to \mathbb{R}_{\not=0}$, is a homomorphism. Show $\det$ is onto. What is its kernel? What does the first isomorphism theorem say here?

\item[\#4] Let $H \triangleleft G$ where $G$ is a finite group. Suppose $xH \in G/H$ and $xH$ is an element of order $m$ in $G/H$. 
                 Show that $G$ must have an element of order $m$. 

{\it Warning:} Just because $xH$ has order $m$ (in $G/H$) does not mean $x$ has order $m$ in $G$.\\ \phantom{\it Warning: } So $x$ itself might not work, but the right {\it power} of $x$ will.

\item[\#5 RESUBMIT] Type up Homework \#7 Problem \#5(a) and its solution in \LaTeX. 

\fbox{\parbox{6.5in}{
Let $G$ and $H$ be groups.\\ 

Show $\{e \} \times H = \{ (e,h) \;|\; h \in H \}$ is a normal subgroup of  $G \times H$ (where $e$ is the identity of $G$).\\

{\it Note:} You need to show that $\{e\} \times H$ is a subgroup AND that it's normal in $G \times H$.
}}

     When typing this problem up, please write carefully: Restate the problem. Write in complete sentences.  
\end{description}

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