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\begin{tabular*}{6in}{c@{\extracolsep{\fill}}c@{\extracolsep{\fill}}c}
  Midterm &\huge TMath 402  &\normalsize Winter 2014\\
\end{tabular*}
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\vskip .1 in

\noindent{True/False: If the statement is false, give a counterexample.\\
 If the statement is \emph{always} true, give a brief explanation of why it is (not \emph{just} an example!).\\  }


\normalsize
\begin{enumerate}

%\item $[4]$ The relation $R$ on the set $\mathbb{Z}$ defined below is an equivalence relation.
%\[R=\{(a,b) \in \mathbb{Z}\times \mathbb{Z} | a-b\geq 0\}.\]
%\vfill 

\item $[4]$ (\S0 \#7) If $a$ and $b$ are positive integers, then $ab=\text{lcm}(a,b)\cdot \gcd(a,b)$.
\vfill

\item $[4]$ (\S1 \#11) Let $a$, $b$, and $c$ be elements in the group $D_4$.  If $ab=bc$ then $a=c$.
\vfill

\item $[4]$ (\S3) Let $G$ be a group and $H$ a subset of $G$ such that $H$ is closed under the group operator.  Then $H$ is a subgroup of $G$.
\vfill
\item $[4]$ (\S2) If $(ab)^2=a^2b^2$ in a group $G$, then $G$ is abelian.
\vfill

\newpage
\item $[9]$ (\S2) For each of the sets and operations below, fill in the entries with yes or no.  \\
If no, briefly explain why.  If yes, briefly describe the process you used to reach that anwer. \\

\hspace{-.8in}\begin{tabular*}{7.3in}{l@{\extracolsep{\fill}}|c |c |c}
\large{Sets $S$ \& Operator $\star$}  	&\large{Forms a Group} &\large{Abelian Group}\\
\hline
\normalsize 							&	&	&\\
a) $\{e, a, b, c \}$						&	&	&\\
$ \begin{matrix}  
\star & e & a & b &c \\
e & e & a & b& c\\
a & a & e & c& b\\
b & b & c & e& a\\
c & c & b & a& e\\
\end{matrix}$				&	&	&\\
							&	&	&\\
\hline

b) $\mathbb{Z}_{8}$						&	&	&\\ %\S1 \#3
$\star$ standard multiplication				&	&	&\\
								&	&	&\\										&	&	&\\										&	&	&\\										&	&	&\\										&	&	&\\

\hline
								&	&	&\\ %\S2 \#10
c) $ \left\{ \left( \begin{matrix}  
a& b \\
c & d\\
\end{matrix}\right) \Bigg\vert ad-bc\not=0 \& a,b,c,d  \in \mathbb{R} \right\}$						&	&	&\\
$\star$ is matrix multiplication					&	&	&\\
								&	&	&\\
								&	&	&\\		
								&	&	&\\										&	&	&\\										&	&	&\\
\hline
								&	&	&\\



\end{tabular*}


\newpage
\large
Free Response: Show your work for the following problems.  The correct answer with 
no supporting work will receive NO credit.\\ \normalsize(If you use a calculator, be sure to tell me.)
\item $[3]$ Describe one application in the world that makes use of one of the mathematical structures discussed thus far in class.
\vspace{.7in}


\item $[8]$ (\S3) Let $G$ be a finite group and consider the Cayley table for $G$. (i.e. the table that shows all possible pairs of binary operators of $G$).  Show that every element of $G$ occurs precisely once in each row of the table.

%\item $[8]$ (\S3) Prove that a group of even order must have an element of order 2.

\vfill
\item   Consider the set $S=\{1,2,3,...n-2,n-1\}$ where $n$ is a positive integer.  
\begin{enumerate}
\item $[2]$ (\S2 \#35) When does $S$ form a group under multiplication?
\vspace{1in}
\item $[6]$ (\S2 \& 0) Prove your assertion.
\vfill
\end{enumerate}


%\vspace{-2.5in}\hspace{1.6in}\includegraphics[height=3in]{blank_5}


% Let $f(x) =\begin{cases}
%			\sqrt{1-(x+3)^2}& \text{if }  -4 \leq x \leq -2\\
%			1   			& \text{if }  -2<x<1\\
%			-(x-2)^2+2 	& \text{if }  1<x\\
%			\end{cases}
%			$
			

\newpage
\item (\S4)  Let $G=\mathbb{Z}_{60}$ under addition modulo $60$.

\begin{enumerate}
\item $[2]$ Find $\vert G \vert$.
\vspace{.25in}

\item $[3]$ Calculate the order of the element $49$ in $G$.
\vspace{1.2in}

\item $[3]$ List four elements of order $15$ in $G$. 
\vfill

\item $[2]$ Find a subgroup of $G$ that has order $5$.
\vspace{.5in} 

\item $[6]$ Write down the subgroup lattice for $\mathbb{Z}_{60}$ 
\vfill
\end{enumerate}





\end{enumerate}

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