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\noindent
\parbox{2.25in}{\bf Math 3110-101}
\hfill {\Large \bf Homework \#7} \hfill
\parbox{2.25in}{\bf \hfill Due: Mon., Nov. $2^\mathrm{nd}$, 2015}

\vspace{0.2in}

\begin{enumerate}
\item Recall that the center of a group is a normal subgroup. $Z = Z(D_6) = \{1,x^3\}$ where $D_6 = \{ 1,x,\dots,x^5,y,xy,\dots,x^5y \}$.
          Find the distinct cosets of $Z$ in $D_6$ then write down a Cayley table for $D_6/Z$. Is $D_6/Z$ abelian? Is it cyclic? Explain your answer.         

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

\item Quotients in $\mathbb{Z}_n$.
   \begin{enumerate}[(a)]
   \item Let $H = \langle 5 \rangle \subseteq \mathbb{Z}_{100}$. First $H= \{ ??? \}$. 
             Then compute the cosets of $H$ in $\mathbb{Z}_{100}$. Write down a Cayley table for $\mathbb{Z}_{100}/H$.
             What familiar group is this isomorphic to?
   \item Let $k,\ell,n$ be positive integers such that $n=k\ell$ (i.e. $k$ divides $n$).
             Make a conjecture about what the quotient $\mathbb{Z}_n / \langle k \rangle$ is isomorphic to. Then prove your conjecture.
             
             {\it Hint:} Define the map $\varphi(x)=x$ from $\mathbb{Z}_n$ to your target group and then use the first isomorphism theorem. Don't forget to
             show that $\varphi$ is a well-defined homomorphism.
   \end{enumerate}

\item Let $G$ be a finite group, $H$ a normal subgroup of $G$, and $g \in G$. Show that $|gH|$ (in the quotient group $G/H$) divides $|g|$ (in $G$).   
         
%\item 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.           
\end{enumerate}



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