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\noindent
\parbox{1.5in}{\bf Math 3110} 
\hfill {\Large \bf  Midterm Exam --- In Class} \hfill
\parbox{1.5in}{\bf \hfill October $14^{\mathrm{th}}$, 2013}

\vspace{0.3in}

\noindent {\large\bf Name:} \underline{\hskip 3.0 truein} \hfill {\bf Be sure to show your work!}

\vspace{0.2in}

\noindent {\bf\large 1. (12 points)} Basics

\begin{enumerate}[(a)]
   \item Let $S = (0,1] = \{ x \;|\; 0 < x \leq 1 \}$. Then $S$ is {\bf not} a group under multiplication. List the group axioms which hold and then using {\bf concrete} counterexamples, show the other axioms fail.

\vspace{0.2in}

Axioms that hold:

\vspace{1.5in}

Axioms that fail:
      
\vspace{1.5in}      

   \item Let $\mathbb{E}$ be the set of even integers. Show $\mathbb{E}$ is a subgroup of $\mathbb{Z}$ using a subgroup test.

\vfill

\end{enumerate}

\newpage
\noindent {\bf\large 2. (14 points)} Cyclic Stuff

   \begin{enumerate}[(a)]

      \item Let $G$ be a finite group and $g \in G$. Suppose that $|g|=30$. 
      
      \begin{enumerate}[i.]
      \item What is the order of $g^{25}$? List the distinct elements in $\langle g^{25} \rangle$.
      
      
      \vspace{1.5in}
      
      
      $\langle g^{25} \rangle = \displaystyle \Bigg\{$ \hspace{4in} $\displaystyle \Bigg\}$
          
      \vspace{0.3in}
      
      \item  Is $g^{904} \in \langle g^{25} \rangle$? \quad {\large\bf Yes \ \  /  \ \  No}
      
      \vspace{1in}
      
      
      \end{enumerate}

     \item How many elements of order 4 does $\mathbb{Z}_{100}$ have? What are they?
     
\vfill

      \item Draw the subgroup lattice for $\mathbb{Z}_{75}$. \quad [{\it Note:} $75 = 3 \cdot 5^2$.]

\vfill


\end{enumerate}

\newpage

\noindent {\bf\large 3. (13 points)} Permutations

\begin{enumerate}[(a)]

\item What is the order of $\sigma = (142)(235)(13)(25)$?

\vspace{1.5in}

\item Let $\sigma = (134)(24)(12)$. Find $\sigma^{-1}$.

\vspace{1.5in}

\item Write $\sigma = (1572)(2345)(12)$ as a product of transpositions. \qquad $\sigma$ is \quad {\large\bf Even \ \  / \ \  Odd}

\vspace{1.5in}

\item Let $\sigma = (12345)(67)$. Compute $\sigma^{995}$.

\vspace{1.5in}

\item Does $S_{9}$ have an element of order 14? If so, give an example. If not, explain why not.

\end{enumerate}

\newpage


\noindent {\bf\large 4. (12 points)} Explain why the following pairs of groups are not isomorphic.

\begin{enumerate}[(a)]

\item $\mathrm{GL}_2(\mathbb{R}) \not\cong \mathbb{C}$

\vfill

\item $U(8) \not\cong \mathbb{Z}_4$

\vfill

\item $A_4 \not\cong D_6$

\vfill

\end{enumerate}


\newpage

\noindent {\bf\large 5. (12 points)} Prove that the following pairs of groups are isomorphic.

\begin{enumerate}[(a)]

\item $U(5) \cong \mathbb{Z}_4$

\vfill

\item Consider $\mathbb{R}_{>0}$ (positive reals under multiplication) and $\mathbb{R}$ (under addition).  Show $\mathbb{R}_{>0} \cong \mathbb{R}$.\\
\force [{\it Hint:} Consider $\varphi(x) = \ln(x)$.]

\vfill

\end{enumerate}


\newpage


\noindent
\force {\bf Math 3110} \hfill {\bf  Midterm Exam --- Take Home} \force

\begin{itemize}
\item This portion of the exam must be turned in {\bf no later} than 4:30pm on Wednesday, October $16^{\mbox{th}}$, 2013.
\item You may use notes, textbooks, and existent online resources to complete these problem. 
\item You may {\bf not} ask anyone (except me and Dr. Vicky Klima) for help.
\end{itemize}

\vspace{0.45in}

\noindent {\bf\large 6. (5 points)} Explain why $111 \in U(997)$. Then compute $111^{-1}$.

\vspace{0.25in}

\noindent {\bf\large 7. (9 points)} Dihedral Problem.

\begin{enumerate}[(a)]

\item Draw a regular hexagon and label the reflective symmetries: $V_1, V_2, \dots, V_6$ (moving around the hexagon in the counter-clockwise direction). Also, let $R_{60^\circ}$ be the counter-clockwise rotation of $60^\circ$. Draw a few pictures to compute $R_{60^\circ} V_1$.

\item Recall that $D_{10} = \langle x,y \;|\; x^{10}=1, y^2=1, (xy)^2=1 \rangle = \{ 1,x,x^2,\dots, x^9, y,xy,x^2y, \dots, x^9y \}$.

Simplify $\alpha = x^4yx^2x^{-3}y^3xy^{-7}$. 

\item Make a table listing the orders of the elements of $D_{24}$ as well as how many elements there are of each order. 

\end{enumerate}

\vspace{0.15in}

\noindent {\bf\large 8. (8 points)} Show $x$ and $gxg^{-1}$ have the same order (in a finite group). Can $x=gxg^{-1}$? If so, what does this say about $g$ and $x$? If not, why not?

\vspace{0.25in}

\noindent {\bf\large 9. (5 points)} Suppose that $\sigma = (13)(24765)$ and your classmate claims that $\tau \sigma \tau^{-1} = (142)(3765)$.   Is this possible? If so, what might $\tau$ be? If not, why not?

\vspace{0.25in}

\noindent {\bf\large 10. (10 points)} Matrices.

\begin{enumerate}[(a)]

\item Let $\displaystyle H = \left\{ \begin{bmatrix} a & b \\ -b & a \end{bmatrix} \;|\; a,b \in \mathbb{R} \mbox{ and } a,b \mbox{ not both 0 } \right\}$. 

\vspace{0.05in}

Using a subgroup test, show that $H$ is a subgroup of $\mathrm{GL}_2(\mathbb{R})$.

\item Prove that $\mathbb{C}_{\not=0} = \{ x+yi \;|\; x,y \in \mathbb{R} \mbox{ and } x,y \mbox{ not both 0 }  \} \cong H$

\end{enumerate}









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