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\noindent
\parbox{1.95in}{\bf Math 3110}
\hfill {\Large \bf Homework \#8} \hfill
\parbox{1.95in}{\bf \hfill Due: Mon., Mar. 31${}^\mathbf{st}$\!, 2025}

%\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] Recall that the center of a group is a normal subgroup. We are given that 
$Z = Z(D_6) = \{1,x^3\}$ where $D_6 = \langle x,y \;|\; x^6=1, y^2=1, (xy)^2=1 \rangle = \{ 1,x,\dots,x^5,y,xy,\dots,x^5y \}$.

          Find the distinct cosets of $Z$ in $D_6$ then write down a Cayley table for $D_6/Z$.\\ \force \hfill Is $D_6/Z$ abelian? Is it cyclic? Explain your answer.         

\item[\#2] Quotients in $\mathbb{Z}_{n}$.
   \begin{enumerate}[(a)]
   \item Let $H = \langle 20 \rangle \subseteq \mathbb{Z}_{50}$. List the elements of $H$ (i.e., $H= \{ ??? \}$). 
             Then compute the cosets of $H$ in $\mathbb{Z}_{50}$. %\sout{Write down a Cayley table for $\mathbb{Z}_{50}/H$.}
             What familiar group is $\mathbb{Z}_{50}/H$ isomorphic to?
   \item Let $k,\ell,n$ be positive integers such that $n=k\ell$ (i.e. $\ell$ divides $n$). It turns out that $\mathbb{Z}_n/ \langle \ell \rangle \cong \mathbb{Z}_\ell$. 
   
   Prove this using the map $\varphi:\mathbb{Z}_n \to \mathbb{Z}_\ell$ defined by $\varphi(x)=x$ and the first isomorphism theorem.
   
   {\it Note:} You need to show $\varphi$ is a {\bf well-defined} homomorphism, then find $\varphi$'s kernel and image, and finally, apply the first isomorphism theorem.
   \end{enumerate}

\item[\#3] Let $\varphi : D_n \to \{ \pm 1\}$ be defined by 
          $\displaystyle \varphi(a) = \left\{ \begin{array}{ccl} +1 & & a \mbox{ is a rotation \Large\strut} \\ -1 & & a \mbox{ is a reflection \Large\strut} \end{array}\right.$
          
          Show $\varphi$ is a homomorphism.  What is the kernel of $\varphi$? What does the first isomorphism theorem tell us here?

\item[\#4] Let $G$ be a finite group, $H$ a normal subgroup of $G$, and $g \in G$. Show that $|gH|$ divides $|g|$ (in $G$).   

{\it Note:} In this problem, $|gH|$ means the order of the element $gH$ in the quotient group $G/H$\\ \force \hfill (as opposed to the cardinality of $gH$ as a set).

{\it Unnecessary Note:} This holds for all groups if one uses the convention:\\ \force \hfill all positive integers as well as infinity itself divides infinity.
         
\item[\#5] Let $G$ and $H$ be finite groups and let $\varphi: G \to H$ be an epimorphism (this is a homomorphism which is onto). 

Suppose that there is some $x \in H$ such that $|x|=8$. Prove that $G$ has an element of order 8 as well.           

\item[RESUBMIT] Type up Homework \#7 Problem \#4(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  $G \times H$\\ \force\hfill (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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