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\begin{tabular*}{6in}{c@{\extracolsep{\fill}}c@{\extracolsep{\fill}}c}
Vanderpool &\huge TMath 402  &\normalsize Autumn 2022\\
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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}

%from DHW1
\item $[3]$ (\S2.2) Let $a$ and $b$ be integers and $p$ be a prime number.  If $p$ divides $ab$, then either $p$ divides $a$ or $p$ divides $b$.
\vfill
\item $[3]$ (\S3.2) All groups are abelian/commutative.
\vfill
%\item $[3]$ (\S4.1) Every subgroup of a cyclic subgroup is cyclic.
%\vfill
\item $[3]$ (\S5.1) Let $\sigma \in S_{12}$. Then $\sigma^{12}=()$.
\vfill







\newpage
\large
Free Response: Show your work for the following problems.  The correct answer with 
no supporting work will receive NO credit.\\ 

\normalsize

\item $[9]$ For each of the sets and operations below, determine if they define a group.  \\
If no, briefly explain why.  If yes, briefly describe the process you used to reach that answer. \\

\hspace{-.8in}\begin{tabular*}{7.8in}{l|@{\extracolsep{\fill}}l@{\extracolsep{\fill}} }
\large{Sets $S$ \& Operator $\star$}  	&\large{Is a Group?} \\
\hline
\normalsize 				
							&		\\
							&		\\
a) $\mathbb{Z}_{10}$ 				&		\\
$\star$ is multiplication under modulo 10	&		\\
							&		\\
							&		\\
\hline
							&		\\
							&		\\
b) $\{a,b,c,d\}$
$\star$ is defined by				&		\\
\begin{tabular*}{1.3in}{|c||c|c|c|c|}
\hline
$\star$ & a & b & c & d\\
\hline
\hline
a & b & a & d & c\\
\hline
b & a & b& c& d\\
\hline
c& c & d &b &a\\
\hline
d& d& c& a& b\\
\hline
\end{tabular*}					&	\\		
							&	\\	
\hline									
							&	\\	
							&\\
c) Set generated by $r$ and $s$ subject 	&		\\
to the relations $e=r^{10}=s^2$ and  & \\
$srs=r^{-1}$ where $e$ is the identity. &\\
							&\\
							&\\
							&\\

\end{tabular*}


%WrittenHW1 and Section 3.1
\item $[2]$ Choose a set (any set you want!) and define a (non-trivial) equivalence relation on it. (Non-trivial means that every element is not in its own equivalence class nor that all elements are in the same equivalence class.)




%\item Dr. Vanderpool provides you with the same Cayley Table for group $W$. Use it to answer the questions below.

%\hspace{3in} \begin{tabular*}{2.6in}{|c||c|c|c|c|c|c|c|c|}
%\hline
%$\star$ &z&x&y&w&r&s&t&u\\
%\hline
%\hline
%z &z&x&y&w&r&s&t&u\\
%\hline
%x &x&r&u&y&s&z&w&t\\
%\hline
%y &y&w&r&s&t&u&z&x\\
%\hline
%w &w&t&x&r&u&y&s&z\\
%\hline
%r &r&s&t&u&z&x&y&w\\
%\hline
%s &s&z&w&t&x&r&u&y\\
%\hline
%t &t&u&z&x&y&w&r&s\\
%\hline
%u &u&y&s&z&w&t&x&r\\
%\hline
%\end{tabular*}






			

\newpage
\item Consider the integers $\mathbb{Z}$ with the the binary operator of addition.  Let $\alpha$ be an integer. Define $\alpha \mathbb{Z}=\{ \alpha k | k\in \mathbb{Z}\}$.

%\vspace{-1.0in} \hspace{3.5in} \includegraphics[height=3.0in]{D3Diagram}


\begin{enumerate}
\item $[4]$ (\S3.3) Show $5\mathbb{Z}$ is a subgroup of $\mathbb{Z}$.
\vspace{3.0in}

\item $[3]$ (ModuloActivity) Draw (a partial) Cayley Diagram for $5\mathbb{Z}$. 
\vspace{1.3in}

\item $[2]$ (4.1) Find a subgroup of $5\mathbb{Z}$.
\vspace{.5in}

\item $[3]$ Create a partial subgroup lattice for $\mathbb{Z}$.  Include $5\mathbb{Z}$ and your answer to part (c) in the lattice.


\end{enumerate}

\newpage
\item $[8]$ Choose \emph{ONE} of the following theorems to prove.  Clearly identify which of the two you are proving and what work you want to be considered for credit.  \\
No, doing both questions will not earn you extra credit.

%Sudoku theorem
\begin{theorem} 
Let $G$ be a group. Prove that $ba=ca$ implies $b=a$ and $ab=ac$ implies $b=c$ for all elements $a$, $b$, and $c$ in $G$.
\end{theorem}

\begin{theorem}
Let $G$ be a finite cyclic group with order $20$ generated by $g$ . Prove $g^n$ generates $G$ if and only if $\gcd(n,20)=1$.  
\end{theorem}

\vfill

\end{enumerate}

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