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\begin{document}

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\noindent
\parbox{2in}{\bf Math 3110} 
\hfill {\Large \bf Final Exam} \hfill
\parbox{2in}{\bf \hfill December 9$^{th}$, 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}/12 points)] For each of the following pairs of groups, if the groups are isomorphic, circle $G_1 \cong G_2$ and explain why they are isomorphic. If the groups aren't isomorphic, circle $G_1 \not\cong G_2$ and explain why not.
\begin{enumerate}[(a)]
   \item \qquad $\mathbb{Q} \cong Q$  \qquad\qquad OR \qquad\qquad $\mathbb{Q} \not\cong Q$ \hfill \hfill [$\mathbb{Q}$ = rationals, $Q$ = quaternions]

\vspace{2in}

   \item \qquad $A_4 \cong D_6$ \qquad\qquad OR \qquad\qquad $A_4 \not\cong D_6$ \hfill \hfill [$A_4$ = even permutations in $S_4$]

\vspace{2in}

   \item \qquad $U(5) \cong \mathbb{Z}_4$ \qquad\qquad OR \qquad\qquad $U(5) \not\cong \mathbb{Z}_4$

\end{enumerate}

\newpage
   \item[\large 2. (\underline{\hskip 0.35 truein}/12 points)] Sub-Things
\begin{enumerate}[(a)]
   \item Let $G$ be a group. Recall that $Z(G) = \{ x \in G \,|\, gx=xg \mbox{ for all } g \in G \}$ is the ``center'' of $G$. Prove that $Z(G)$ is a normal subgroup of $G$. [Hint: You may either {\it show it is a subgroup} {\bf and} {\it show it is normal}, or you can find a homomorphism with kernel $Z(G)$.]

\vspace{3in}

   \item Let $S = \{ m+n\sqrt{5} \,|\, m,n \in \mathbb{Z} \}$. Show $S$ is a sub{\bf{}ring} of $\mathbb{R}$.

\vspace{3in}

   \item Let $T = \{ m+n\sqrt{5} \,|\, m,n \in \mathbb{Z} \mbox{ and } m \mbox{ is \bf EVEN}\}$. Show $T$ is a {\bf subgroup} of $\mathbb{R}$ (under addition of course) and then give a concrete counter-example which shows why $T$ is {\bf not} a sub{\bf{}ring} of $\mathbb{R}$.

\end{enumerate}

\newpage
   \item[\large 3. (\underline{\hskip 0.35 truein}/13 points)] Calculatin' mod 50. \quad [${\bf 50 = 2 \cdot 5^2}$]
\begin{enumerate}[(a)]
   \item Is $3$ a unit, zero divisor, or neither in $\mathbb{Z}_{50}$?
If $3$ is a unit, find its inverse. If $3$ is zero divisor, show this by finding  some $0 \not= m \in \mathbb{Z}_{50}$ such that $3m=0$ (mod 50).

\vspace{2in}

   \item Is $20$ a unit, zero divisor, or neither in $\mathbb{Z}_{50}$?
If $20$ is a unit, find its inverse. If $20$ is zero divisor, show this by finding some $0 \not= m \in \mathbb{Z}_{50}$ such that $20m=0$ (mod 50).

\vspace{2in}

\item Find all of the principle ideals of $\mathbb{Z}_{50}$.

\vspace{2in}

\item How many elements generate $\mathbb{Z}_{50}$?\\ {}[That is: How
many $x \in \mathbb{Z}_{50}$ are there such that $(x)=\langle x \rangle = \mathbb{Z}_{50}$?] 

\end{enumerate}

\newpage
   \item[\large 4. (\underline{\hskip 0.35 truein}/15 points)] The set $I = (4) = \{ 0,4,8,12,16 \}$ is an ideal of $\mathbb{Z}_{20}$. 
   Let $\displaystyle{R = {\mathbb{Z}_{20} \atop \;}\hspace*{-0.1in} \mbox{\put(0,0){\line(2,1){2}}} \hspace*{0.1in} {\; \atop I}}$
\begin{enumerate}[(a)]
   \item Write down all of the distinct cosets of $I$ in $\mathbb{Z}_{20}$. 

\vspace{1in}


   \item Finish filling out the following addition and multiplication tables for $R$:

\vspace{0.1in}

\noindent
\mbox{} \hspace{-0.6in}
\begin{tabular}{|c||c|c|c|c|} 
\hline
 $+$ &  \varspace{0.3in}{0.2in} + I &  \varspace{0.3in}{0.2in} + I & \varspace{0.3in}{0.2in} + I & \varspace{0.3in}{0.2in} + I \\ \hline \hline
\varspace{0.3in}{0.2in} + I &  \varspace{0.3in}{0.2in} + I & \varspace{0.3in}{0.2in} + I & \varspace{0.3in}{0.2in} + I & \varspace{0.3in}{0.2in} + I \\ \hline
\varspace{0.3in}{0.2in} + I &  \varspace{0.3in}{0.2in} + I & \varspace{0.3in}{0.2in} + I & \varspace{0.3in}{0.2in} + I & \varspace{0.3in}{0.2in} + I \\ \hline
\varspace{0.3in}{0.2in} + I &  \varspace{0.3in}{0.2in} + I & \varspace{0.3in}{0.2in} + I & \varspace{0.3in}{0.2in} + I & \varspace{0.3in}{0.2in} + I \\ \hline
\varspace{0.3in}{0.2in} + I &  \varspace{0.3in}{0.2in} + I & \varspace{0.3in}{0.2in} + I & \varspace{0.3in}{0.2in} + I & \varspace{0.3in}{0.2in} + I \\ \hline
\end{tabular} 
\hspace{0.2in}
\begin{tabular}{|c||c|c|c|c|} 
\hline
 $\times$ &  \varspace{0.3in}{0.2in} + I &  \varspace{0.3in}{0.2in} + I & \varspace{0.3in}{0.2in} + I & \varspace{0.3in}{0.2in} + I \\ \hline \hline
\varspace{0.3in}{0.2in} + I &  \varspace{0.3in}{0.2in} + I & \varspace{0.3in}{0.2in} + I & \varspace{0.3in}{0.2in} + I & \varspace{0.3in}{0.2in} + I \\ \hline
\varspace{0.3in}{0.2in} + I &  \varspace{0.3in}{0.2in} + I & \varspace{0.3in}{0.2in} + I & \varspace{0.3in}{0.2in} + I & \varspace{0.3in}{0.2in} + I \\ \hline
\varspace{0.3in}{0.2in} + I &  \varspace{0.3in}{0.2in} + I & \varspace{0.3in}{0.2in} + I & \varspace{0.3in}{0.2in} + I & \varspace{0.3in}{0.2in} + I \\ \hline
\varspace{0.3in}{0.2in} + I &  \varspace{0.3in}{0.2in} + I & \varspace{0.3in}{0.2in} + I & \varspace{0.3in}{0.2in} + I & \varspace{0.3in}{0.2in} + I \\ \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
$R$ has a ``unity''                   & & \varspace{0.5in}{3.5in} \\ \hline
$R$ is commutative                    & & \varspace{0.5in}{3.5in} \\ \hline
$R$ is an integral domain             & & \varspace{0.5in}{3.5in} \\ \hline
$R$ is a field                        & & \varspace{0.5in}{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{-0.65in}
\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} & \varspace{0.35in}{1.75in} \\ \hline
\myspace & \myspace \\ \hline
\myspace & \myspace \\ \hline
\myspace & \myspace \\ \hline
\end{tabular}
\hspace{0.2in}
\begin{tabular}{|c|c|} \hline
Units & \parbox[c][0.4in]{1.25in}{My multiplicative\\ inverse is...} \\ \hline \hline
\parbox[c]{1in}{\hfill} & \varspace{0.35in}{1.75in} \\ \hline
\myspace & \myspace \\ \hline
\myspace & \myspace \\ \hline
\myspace & \myspace \\ \hline
\end{tabular}

\end{enumerate}

\newpage
   \item[\large 5. (\underline{\hskip 0.35 truein}/12 points)] Homomorphisms.
\begin{enumerate}[(a)]
   \item Show $\varphi : \mathbb{Z}_3 \rightarrow \mathbb{Z}_6$ defined
by $\varphi(x)=2x$ is a {\bf well-defined} homomorphism.

\vspace{2in}

   \item Find the kernel and image of $\varphi$. Is $\varphi$ one-to-one? onto? an isomorphism?

\vspace{1.5in}

   \item $\displaystyle{S = \left\{ \begin{bmatrix} a & 0 \\ 0 & b \end{bmatrix} \,\Big|\, a,b \in \mathbb{R} \right\}}$ is a subring of $M_2(\mathbb{R}) = \mathbb{R}^{2 \times 2}$. Define $\psi : S \rightarrow \mathbb{R}$ by
$\displaystyle{\psi \left(\begin{bmatrix} a & 0 \\ 0 & b \end{bmatrix} \right) = a}$. Show that $\psi$ is a ring homomorphism.

\vspace{2in}

   \item Find the kernel and image of $\psi$. Is $\psi$ one-to-one? onto? an isomorphism?


\end{enumerate}

\newpage
   \item[\large 6. (\underline{\hskip 0.35 truein}/12 points)] Some random proofs.
\begin{enumerate}[(a)]
   \item Let $R$ be a ring with $1$. Let $u$ be a unit of $R$. Show that $u$ cannot be a zero divisor. (This shows that all fields are integral domains.)

\vspace{2.5in}

   \item Let $R$ be a commutative ring with $1$. Suppose $a \in R$ and recall that $(a) = \{ ra \,|\, r \in R \}$ is the principle ideal generated by $a$. Prove that $(a)$ is an ideal of $R$.

\vspace{3.5in}

   \item Let $G$ be a group such that $(xy)^{-1}=x^{-1}y^{-1}$ for all $x,y \in G$. Show that $G$ is abelian.

\end{enumerate}

\newpage
   \item[\large 7. (\underline{\hskip 0.35 truein}/12 points)] Just a little harmless permuting.
\begin{enumerate}[(a)]
   \item Let $H = \{ (1),(12) \}$. Find all of the {\bf left and right} cosets of $H$ in $S_3 = \{ (1),(12),(13),(23),(123),(132) \}$. Is $H$ a normal subgroup?

\vspace{3in}

   \item Explain why $A_n$ (the even permutations) is a normal subgroup of $S_n$. [Hint: You may either briefly explain {\it why it is a subgroup} {\bf and} {\it why it is normal}, or you can find a homomorphism with kernel $A_n$.]

\vspace{3in}

   \item Consider the quotient group $\displaystyle{{S_n \atop \;}\hspace*{-0.1in} \mbox{\put(0,0){\line(2,1){2}}} \hspace*{0.1in} {\; \atop A_n}}$. It this a cyclic group? Why or why not?

\end{enumerate}

\newpage
   \item[\large 8. (\underline{\hskip 0.35 truein}/12 points)] $D_{10}$. $n=10$ really? why oh why?
\begin{enumerate}[(a)]
   \item Let $G$ be a {\bf non-abelian} group of order 20. What are possible orders of elements of $G$? What does ``non-abelian'' rule out and why?

\vspace{2in}

   \item Half of the elements of $D_{10}$ are reflections and half are rotations. In fact, the {\bf rotations} in $D_{10}$ form a {\bf cyclic subgroup}.
Without worrying about the cyclic part, {\bf briefly} explain why the rotations do form a subgroup and why the reflections do not.

\vspace{2in}

   \item Use the description of $D_{10}$ given in part (b) to determine the number of elements of each order in $D_{10}$.

\vspace{0.1in}

{} \hspace*{-0.5in}
\begin{tabular}{r}
Element order = \\
{} \\
Number of elements with that order = \\
\end{tabular}

\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}
