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\noindent
\parbox{2in}{\bf Math 3110} 
\hfill {\Large \bf Test \#3} \hfill
\parbox{2in}{\bf \hfill November 20$^{th}$, 2009}

\vspace{0.3in}

\noindent {\large\bf Name:} \underline{\hskip 3.0 truein}\\
Don't merely state answers, prove your statements. {\bf Be sure to show your work!}

\noindent
{\LARGE \bf The last page is a copy of Cayley tables for $D_4$ and $Q$.} 

\begin{description}
   \item[\large 1. (\underline{\hskip 0.35 truein}/16 points)] The following groups are {\bf NOT} isomorphic. Explain why.
\begin{enumerate}[(a)]
   \item $D_5 \not\cong \mathbb{Z}_{10}$

\vspace{1in}

   \item $D_4 \not\cong Q$ where $Q = \{ \pm 1, \pm i, \pm j, \pm k \}$ (the quaternions).

\vspace{1.5in}

\end{enumerate}

   \item[\large 2. (\underline{\hskip 0.35 truein}/14 points)] Is it possible?
\begin{enumerate}[(a)]
   \item Let $G_1$ and $G_2$ be groups of order 7. Is it possible that $G_1 \not\cong G_2$? If so,
            give examples of non-isomorphic groups of order 7. If $G_1$ is always isomorphic to $G_2$, explain why.

\vspace{2in}

   \item Let $G_1$ and $G_2$ be groups of order 6. Is it possible that $G_1 \not\cong G_2$? If so,
            give examples of non-isomorphic groups of order 6. If $G_1$ is always isomorphic to $G_2$, explain why.

\end{enumerate}


\newpage
   \item[\large 3. (\underline{\hskip 0.35 truein}/18 points)] Homomorphisms
\begin{enumerate}[(a)]
      \item Quickly, explain why $\psi : \mathbb{Z}_3 \rightarrow \mathbb{Z}_7$ defined by $\psi(x)=x$
                is {\bf NOT} a well-defined function.
                
     \vspace{1in}
     
     \item Let $\varphi : \mathbb{Z}_8 \rightarrow \mathbb{Z}_{12}$ be defined by $\varphi(x)=6x$. Show that $\varphi$ is
               a well-defined homomorphism\\ (i.e. show that $\varphi$ is well-defined {\it and} $\varphi$ is a homomorphism).
      
    \vspace{3in}
    
    \item Using the homomorphism $\varphi$ from part (b), find $\mbox{Ker}(\varphi)$ and $\varphi(\mathbb{Z}_8)$.
    
    \vspace{0.2in}
    
              Is $\varphi$ one-to-one? \underline{\hskip 0.5 truein}  \qquad
              Is $\varphi$ onto? \underline{\hskip 0.5 truein}  \qquad
              Is $\varphi$ an isomorphism? \underline{\hskip 0.5 truein}  
      
\end{enumerate}

\newpage
   \item[\large 4. (\underline{\hskip 0.35 truein}/16 points)] Write down the left
   multiplication operators for $U(8) = \{ 1,5,3,7 \}$ then translate them into permuations (in $S_4$)
   labeling the elements of $U(8)$ as follows: 
   \[ 1 \rightsquigarrow 1, \qquad 5 \rightsquigarrow 2, \qquad 3 \rightsquigarrow 3, \qquad \mbox{and} \qquad 7 \rightsquigarrow 4. \]
   Finally, write down a subgroup of $S_4$ which isomorphic to $U(8)$ (using Cayley's theorem).

\newpage
   \item[\large 5. (\underline{\hskip 0.35 truein}/18 points)] Quotient Groups
\begin{enumerate}[(a)]
      \item Let $H = \{0,4,8\} \lhd \mathbb{Z}_{12}$. Write out all of the cosets of $H$. \\ Then fill out a Cayley table for the quotient
                $\displaystyle{{\mathbb{Z}_{12} \atop \;}\hspace*{-0.1in} \mbox{\put(0,0){\line(2,1){2}}} \hspace*{0.1in} {\; \atop H}}$.

\vspace{4in}

      \item Consider $K = \{ 1, x^2 \}$ in $D_4$. Then $xK \in \displaystyle{{D_4 \atop \;}\hspace*{-0.1in} \mbox{\put(0,0){\line(2,1){2}}} \hspace*{0.1in} {\; \atop K}}$.\\ Compute $(xK)^2 = xK \, xK$. What is the order of $xK$ (as an element of this quotient group)?
      
      
\end{enumerate}

\newpage
   \item[\large 6. (\underline{\hskip 0.35 truein}/18 points)] Other stuff.
\begin{enumerate}[(a)]
      \item Let $H$ be a subgroup of $G$ and $K$ be a subgroup of $H$ ($K \stackrel{\subseteq\,}{\mbox{\tiny s.g.}} H
      \stackrel{\subseteq\,}{\mbox{\tiny s.g.}} G$).
      
      \vspace{0.1in}
      
       If $|G|=18$, then what are the possible orders of $H$? \underline{\hskip 2.5 truein}
      
      \vspace{0.05in}
      
      Suppose that $H \not= G$ and $H$ has more than 7 elements. Also, $K$ contains more than just the identity. What are the possible orders
      of $K$?\\ {}[Hint: determine the order of $H$ first, then remember that $K$ is a subgroup of $H$.]
      
\vspace{3in}

      \item Let $G$ be an {\bf Abelian} group and $H \stackrel{\subseteq\,}{\mbox{\tiny s.g.}} G$. Give a quick explanation why $H$ is
      {\bf normal} in $G$ and then prove that $\displaystyle{{G \atop \;}\hspace*{-0.1in} \mbox{\put(0,0){\line(2,1){2}}} \hspace*{0.1in} {\; \atop H}}$ is Abelian.  

\end{enumerate}

\end{description}

\newpage
\LARGE
\begin{tabular}{|c|||c|c|c|c|c|c|c|c|}
\hline
$D_4$  & $1$    & $x$    & $x^2$  & $x^3$  & $y$    & $xy$   & $x^2y$ & $x^3y$ \\ \hline 
\hline \hline
$1$    & $1$    & $x$    & $x^2$  & $x^3$  & $y$    & $xy$   & $x^2y$ & $x^3y$ \\ \hline 
$x$    & $x$    & $x^2 $ & $x^3 $ & $1   $ & $x  y$ & $x^2y$ & $x^3y$ & $   y$ \\ \hline 
$x^2$  & $x^2$  & $x^3 $ & $1   $ & $x   $ & $x^2y$ & $x^3y$ & $   y$ & $x  y$ \\ \hline 
$x^3$  & $x^3$  & $1   $ & $x   $ & $x^2 $ & $x^3y$ & $   y$ & $x  y$ & $x^2y$ \\ \hline 
$y$    & $y$    & $x^3y$ & $x^2y$ & $x  y$ & $1   $ & $x^3 $ & $x^2 $ & $x   $ \\ \hline 
$xy$   & $xy$   & $   y$ & $x^3y$ & $x^2y$ & $x   $ & $1   $ & $x^3 $ & $x^2 $ \\ \hline 
$x^2y$ & $x^2y$ & $x  y$ & $   y$ & $x^3y$ & $x^2 $ & $x   $ & $1   $ & $x^3 $ \\ \hline 
$x^3y$ & $x^3y$ & $x^2y$ & $x  y$ & $   y$ & $x^3 $ & $x^2 $ & $x   $ & $1   $ \\ \hline 
\end{tabular}

\vspace{1in}

\begin{tabular}{|r|||r|r|r|r|r|r|r|r|}
\hline
$Q$  & $1$    & $-1$    & $i$       & $-i$      & $j$    & $-j$      & $k$       & $-k$ \\ \hline 
\hline \hline
$1$  & $1$    & $-1$    & $i$       & $-i$      & $j$    & $-j$      & $k$       & $-k$ \\ \hline 
$-1$ & $-1$   & $1$     & $-i$      & $i$       & $-j$   & $j$       & $-k$      & $k$  \\ \hline 
$i$   & $i$      & $-i$     & $-1$     & $1$      & $k$    & $-k$    & $-j$       & $j$ \\ \hline 
$-i$  & $-i$     & $i$      & $1$      & $-1$     & $-k$   & $k$     & $j$       & $-j$ \\ \hline 
$j$   & $j$      & $-j$     & $-k$     & $k$      & $-1$    & $1$    & $i$       & $-i$ \\ \hline 
$-j$  & $-j$     & $j$      & $k$      & $-k$     & $1$    & $-1$     & $-i$       & $i$ \\ \hline 
$k$   & $k$      & $-k$     & $j$     & $-j$      & $-i$    & $i$    & $-1$       & $1$ \\ \hline 
$-k$  & $-k$     & $k$      & $-j$   & $j$        & $i$   & $-i$     & $1$       & $-1$ \\ \hline 
\end{tabular}

\end{document}
