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\noindent
\parbox{2in}{\bf Math 3110} 
\hfill {\Large \bf Test \#2} \hfill
\parbox{2in}{\bf \hfill March 25$^{th}$, 2009}

\vspace{0.3in}

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

\begin{description}
   \item[\large 1. (\underline{\hskip 0.35 truein}/16 points)] Either prove $G$ is a group
        or explain why it is not a group.
\begin{enumerate}[(a)]
      \item $G = 5\mathbb{Z} = \{ m \in \mathbb{Z} \,|\, m \;\mathrm{is\;divisible\;by}\;5\}$
            (under addition).

\vspace{4in}

      \item $G = (0,1] = \{ r \in \mathbb{R} \,|\, 0<r\leq 1 \}$ (under multiplication).

   \end{enumerate}

\newpage
   \item[\large 2. (\underline{\hskip 0.35 truein}/20 points)] Subgroups
\begin{enumerate}[(a)]
      \item Show $\displaystyle{ H = \left\{ \begin{bmatrix} 1 & a \\ 0 & 1 \end{bmatrix} \,\Bigg|\, a \in \mathbb{R} \right\}}$ is a subgroup of $\mathrm{GL}_2(\mathbb{R})$ 

\vspace{0.1in}

(Recall $\mathrm{GL}_2(\mathbb{R})$ --- the set of $2 \times 2$ invertible matrices ---
 is a group under matrix multiplication).

\vspace{6in}

      \item Informally (in a sentence or two) explain why $R = \{ x \in D_n \,|\, x \;\mathrm{is\;a\;rotation} \}$ is a subgroup of $D_n$.

   \end{enumerate}

\newpage
   \item[\large 3. (\underline{\hskip 0.35 truein}/20 points)] Quick proofs
\begin{enumerate}[(a)]
      \item Show that Cyclic implies Abelian.

\vspace{4in}

      \item Suppose that $G$ is a group such that $xyx^{-1}y^{-1}=1$ for all $x,y \in G$. 
            Show that $G$ is Abelian.

   \end{enumerate}

\newpage
   \item[\large 4. (\underline{\hskip 0.35 truein}/20 points)] Calculatin' mod $n$
\begin{enumerate}[(a)]
      \item Find the order of each of the elements of $\mathbb{Z}_6$. Find all of the
            subgroups of $\mathbb{Z}_6$ and draw a subgroup lattice. Briefly explain how
            you know that you've found all of the subgroups.

\vspace{6in}

      \item Write a Cayley table for $U(8)$. Determine the order of each element of $U(8)$.
            {\bf Is $U(8)$ cyclic?}

   \end{enumerate}

\newpage
   \item[\large 5. (\underline{\hskip 0.35 truein}/24 points)] Permutations! 
\begin{enumerate}[(a)]
      \item Write the order of each of the following permutations below the permutation.\\

\vspace{0.1in}

$S_4 = \{ (1), \;\; 
          (12), \;\; (13), \;\; (14), \;\; (23), \;\; (24), \;\; (34), \;\; 
          (123), \;\; (132), \;\; (124), \;\; (142), \;\; (134), \;\; (143),$

\vspace{0.4in}

$ \qquad (234), \;\; (243), \;\; (1234), \;\; (1243), \;\; (1324), \;\; (1342), \;\; (1423), 
  \;\; (1432), \;\; (12)(34), \;\; (13)(24), \;\; (14)(23) \}$

\vspace{0.4in}

      \item Write $(1432)$ as a product of transpositions and find $\langle\, (1432)\, \rangle$
            (the subgroup generated by $(1432)$).

\vspace{2.5in}

      \item List the elements of $A_4$ (the subgroup of even permutations). 
            {\bf Is $A_4$ cyclic?}

\vspace{3in}

      \item Write $\sigma = (1243)(526)(1946)(78)$ as a product of disjoint cycles. Then
            find the order of $\sigma$.
            

   \end{enumerate}
\end{description}

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