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\noindent
\parbox{2in}{\bf Math 3110} 
\hfill {\Large \bf Final Exam} \hfill
\parbox{2in}{\bf \hfill May 4$^{th}$, 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
\begin{description}
   \item[\large 1. (\underline{\hskip 0.35 truein}/12 points)] Tables for Sarah. Let 
        $R = \{ 0,3,6,9 \} \subset \mathbb{Z}_{12}$.
\begin{enumerate}[(a)]
   \item Finish filling out the following addition and multiplication tables for $R$:

\vspace{0.1in}

\noindent
\mbox{} \hspace{0.1in}
\begin{tabular}{|c||c|c|c|c|} 
\hline
 $+$ &        0 &        3 &        6 &        9 \\ \hline \hline
   0 &        0 &        3 &        6 &        9 \\ \hline 
   3 &        3 & \myspace & \myspace & \myspace \\ \hline 
   6 &        6 & \myspace & \myspace & \myspace \\ \hline 
   9 &        9 & \myspace & \myspace & \myspace \\ \hline 
\end{tabular} 
\hspace{0.2in}
\begin{tabular}{|c||c|c|c|c|} 
\hline
 $\times$ &        0 &        3 &        6 &        9 \\ \hline \hline
        0 &        0 &        0 &        0 &        0 \\ \hline 
        3 &        0 & \myspace & \myspace & \myspace \\ \hline 
        6 &        0 & \myspace & \myspace & \myspace \\ \hline 
        9 &        0 & \myspace & \myspace & \myspace \\ \hline 
\end{tabular}

\vspace{0.1in}

   \item Fill out the following table of information about $R$.

\vspace{0.1in}

\noindent \mbox{} \hspace{-0.65in}
\begin{tabular}{|c|c|c|}
\hline
Property & \parbox[c][0.5in]{0.75in}{Is this true\\ about $R$?} & Briefly Explain \\ \hline \hline
$R$ is a subring of $\mathbb{Z}_{12}$ & & \varspace{0.65in}{3.5in} \\ \hline
$R$ has a ``unity''                   & & \varspace{0.65in}{3.5in} \\ \hline
$R$ is commutative                    & & \varspace{0.65in}{3.5in} \\ \hline
$R$ is an integral domain             & & \varspace{0.65in}{3.5in} \\ \hline
$R$ is a field                        & & \varspace{0.65in}{3.5in} \\ \hline
\end{tabular}

\vspace{0.1in}

   \item Fill out the following table concerning $R$. 
         {\it Hint:} These tables are longer than they need to be.

\vspace{0.1in}

\mbox{} \hspace{-0.25in}
\begin{tabular}{|c|c|} \hline
Zero Divisors & \parbox[c][0.4in]{1.25in}{I am a zero divisor\\ because...} \\ \hline \hline
\parbox[c]{1in}{\hfill} & \myspace \\ \hline
\myspace & \myspace \\ \hline
\myspace & \myspace \\ \hline
\myspace & \myspace \\ \hline
\end{tabular}
\hspace{0.5in}
\begin{tabular}{|c|c|} \hline
Units & \parbox[c][0.4in]{1.25in}{My multiplicative\\ inverse is...} \\ \hline \hline
\parbox[c]{1in}{\hfill} & \myspace \\ \hline
\myspace & \myspace \\ \hline
\myspace & \myspace \\ \hline
\myspace & \myspace \\ \hline
\end{tabular}

\end{enumerate}

\newpage
\item[\large 2. (\underline{\hskip 0.35 truein}/13 points)] 
   Recall $A_4 = \{ (1), (123), (132), (124), (142), (134), (143), (234), (243),$\\
                   $(12)(34), (13)(24), (14)(23) \}$. 
   Let $H = \{ (1), (12)(34), (13)(24), (14)(23) \}$.
\begin{enumerate}[(a)]
   \item Quickly explain why $H$ is a subgroup of $A_4$ (I am thinking of a certain 
         one word answer). 

\vspace{0.5in}
 
   \item Find all of the left and right cosets of $H$ in $A_4$. Is $H$ a {\bf normal} subgroup
         of $A_4$?

\vspace{4.25in}

   \item Construct a Cayley table for $\displaystyle{{A_4 \atop \;}\hspace*{-0.1in} 
         \mbox{\put(0,0){\line(2,1){2}}} \hspace*{0.1in} {\; \atop H}}$.

\vspace{2.75in}

   \item Is $\displaystyle{{A_4 \atop \;}\hspace*{-0.1in} 
         \mbox{\put(0,0){\line(2,1){2}}} \hspace*{0.1in} {\; \atop H}}$ Abelian? Cyclic?
         Why or why not?
\end{enumerate}

\newpage
\item[\large 3. (\underline{\hskip 0.35 truein}/13 points)] An ``ideal'' exam question.
\begin{enumerate}[(a)]
   \item Find all of the principle ideals of $\mathbb{Z}_{20}$. {\it Hint:}
         Principle ideals of $\mathbb{Z}_n$ = Cyclic subgroups of $\mathbb{Z}_n$.

\vspace{3.5in}

   \item It turns out that $\mathbb{Z}_{15}$ is a principle ideal of itself. Find
         all $k \in \mathbb{Z}_{15}$ such that $(k)=\mathbb{Z}_{15}$.\\ In general,
         what do we call the set of all $k \in \mathbb{Z}_n$ such that $(k)=\mathbb{Z}_n$?\\
         {\it Hint:} Principle ideals of $\mathbb{Z}_n$ = cyclic subgroups in $\mathbb{Z}_n$, 
         so I'm really asking you about the generators of the group $\mathbb{Z}_n$.

\vspace{2.5in}

   \item Let $R$ be a ring with $1$, let $I$ be an ideal of $R$, and suppose $u \in R$ is
         a unit.\\ Prove that $u \in I$ implies that $I=R$.
\end{enumerate}

\newpage
\item[\large 4. (\underline{\hskip 0.35 truein}/14 points)] Random Questions.
\begin{enumerate}[(a)]
   \item Consider $22 \in \mathbb{Z}_{125}$. Is $22$ a unit, zero divisor, both, or neither?\\
         If $22$ is a unit, find its inverse. If $22$ is a zero divisor find $b \not=0$ such
         that $22b = 0 \mbox{ mod } 125$.

\vspace{2.5in}

   \item Let $G$ be an Abelian group and $H$ a subgroup of $G$. Prove that $H$ is a normal
         subgroup of $G$.

\vspace{1.5in}

   \item Let $G$ be a cyclic group and let $H$ be a subgroup of $G$. Quickly explain why 
         $\displaystyle{{G \atop \;}\hspace*{-0.1in} \mbox{\put(0,0){\line(2,1){2}}}
          \hspace*{0.1in} {\; \atop H}}$ is a group.\\ (i.e. Why can we quotient by $H$?) 
         Then prove that 
         $\displaystyle{{G \atop \;}\hspace*{-0.1in} \mbox{\put(0,0){\line(2,1){2}}}
          \hspace*{0.1in} {\; \atop H}}$ is cyclic.
  
\end{enumerate}

\newpage
\item[\large 5. (\underline{\hskip 0.35 truein}/12 points)] Isomorphic or not?
\begin{enumerate}[(a)]
   \item $\mathbb{R}_{>0}$ is the group of positive real numbers under multiplication.
         $\mathrm{GL}_2(\mathbb{R})$ is the group of invertible $2 \times 2$ real
         matrices. Is $\mathbb{R}_{>0} \cong \mathrm{GL}_2(\mathbb{R})$? Why or why not?

\vspace{2.5in}

   \item Is $U(10) \cong \mathbb{Z}_4$? Why or why not?

\vspace{2.5in}

   \item Is $S_4 \cong D_{12}$? Why or why not? 
\end{enumerate}

\newpage
\item[\large 6. (\underline{\hskip 0.35 truein}/13 points)] Subgroups
\begin{enumerate}[(a)]
   \item Show that $K = \{ \ell \in \mathbb{Z} \,|\, 4 \mbox{ divides } \ell \}$ 
         is a subgroup of $\mathbb{Z}$.

\vspace{3in}

   \item Let $G$ be an Abelian group with identity $e$. 
         Let $H = \{ g \in G \,|\, g^2=e \}$.\\ Show that $H$ is a subgroup of $G$.

\vspace{3in}

   \item Let $N = \{ x \in \mathbb{Z}_{99} \,|\, 5x \equiv 2 \mbox{ mod } 99 \}$.
         Is $N$ a subgroup of $\mathbb{Z}_{99}$? Why or why not?
\end{enumerate}

\newpage
\item[\large 7. (\underline{\hskip 0.35 truein}/14 points)] Subrings
\begin{enumerate}[(a)]
   \item Let $\displaystyle{ S = \left\{ \begin{bmatrix} a & b \\ 0 & c \end{bmatrix}
   \,\Big|\, a,b,c \in \mathbb{R} \right\}}$ is a subring of 
   $M_2(\mathbb{R}) = \mathbb{R}^{2 \times 2}$ (the ring of $2 \times 2$ real matrices).\\
   Show $S$ is a subring of $M_2(\mathbb{R})$.

\vspace{4in}

   \item Let $R$ be a ring and $a \in R$. Show that $T = \{ r \in R \,|\, ar = 0\}$ 
         is a subring of $R$.

\vspace{3in}

   \item Let $T$ be the subring of $R$ from part (b). If $a=0$, what is $T$?

\vspace{0.6in}

   \item Let $T$ be the subring of $R$ from part (b). If $R$ is an integral domain 
         and $a \not=0$, what is $T$?
\end{enumerate}

\newpage
\item[\large 8. (\underline{\hskip 0.35 truein}/15 points)] Homomorphisms and Kernels.

\vspace{0.1in}

Let $\mathbb{R}^{\#}$ be the group of non-zero real numbers under multiplication.

\noindent
\begin{enumerate}[(a)]
   \item Let $f : \mathbb{R}^{\#} \rightarrow \mathbb{R}$ be defined by $f(x) = \ln|x|$
         (notice the absolute value sign around $x$).\\
         Show that $f$ is a homomorphism. Find $\mathrm{Ker}(f)$. Is $f$ 1-1? onto? an
         isomorphism?

\vspace{2in}

   \item Let $G$ be a group and $x,g \in G$. Let $\varphi_g(x) = gxg^{-1}$. Recall
         that $\varphi_g$ is an automorphism of $G$ (that is $\varphi_g \in \mbox{Aut}(G)$).
         Also, recall that $\mbox{Aut}(G)$ (automorphisms of $G$) is a group (under 
         function composition). 
         Consider the map $\psi:G \rightarrow \mbox{Aut}(G)$ defined by $\psi(g) = \varphi_g$.
         
   \begin{enumerate}[(i)]
      \item Show that $\varphi_g \circ \varphi_h = \varphi_{gh}$.\\ 
            (This shows that $\psi$ is a homomorphism.)

\vspace{1.75in}

      \item Show that $\mbox{Ker}(\psi) = Z(G) = \{ g \in G \,|\, gx=xg \; \mbox{ for all }
            x \in G\}$\\ (Therefore, the center of $G$ is a normal subgroup).

\vspace{2.25in}

      \item Let $\mbox{Inn}(G) = \{ \varphi_g \,|\, g \in G\}$ (``inner automorphisms'' of 
            $G$).\\ Use part (ii) to explain why $\displaystyle{{G \atop \;}\hspace*{-0.1in} 
         \mbox{\put(0,0){\line(2,1){2}}} \hspace*{0.1in} {\; \atop Z(G)}} \cong \mbox{Inn}(G)$.
   \end{enumerate}
\end{enumerate}

\end{description}

\end{document}