\documentclass{article}
\usepackage{graphicx}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{enumerate}
\usepackage{ifthen}
\usepackage{ulem} % for strikethrough \sout{...} command
\usepackage{color}

\usepackage{wasysym}

\usepackage{hyperref}

\setlength{\unitlength}{0.1in}

\setlength{\oddsidemargin}{-0.25in}
\setlength{\evensidemargin}{-0.25in}
\setlength{\topmargin}{-0.25in}
\setlength{\headheight}{0.0in}
\setlength{\headsep}{0.0in}
\setlength{\topskip}{0.0in}
\setlength{\textheight}{9.75in}
\setlength{\textwidth}{7in}

\newcommand{\bdot}{\, {\scriptstyle{\bullet}} \,}
\newcommand{\B}[1]{{\bf {#1}}}
\newcommand{\la}{\langle}
\newcommand{\ra}{\rangle}

\newcommand{\myspace}{\parbox[c][0.25in]{0.25in}{\hfill}}
\newcommand{\varspace}[2]{\mbox{\parbox[c][#1]{#2}{\hfill}}}
\newcommand{\force}{{\noindent $\mbox{\;}$}}

\newcommand{\diagquotient}[2]{\displaystyle{{#1 \atop \;}\hspace*{-0.1in} \mbox{\put(0,0){\line(2,1){2}}} \hspace*{0.1in} {\; \atop #2}}}

\begin{document}

\pagestyle{empty}

\noindent
\parbox{2.25in}{\bf Math 3110}
\hfill {\Large \bf Homework \#4} \hfill
\parbox{2.25in}{\bf \hfill Due: \sout{Fri., Sept. 27${}^\mathbf{th}$, 2024}}

%\vspace{0.2in}
%
%\noindent {\bf\large Name:} \quad \underline{\large\qquad This is my Name \qquad}

\vspace{0.05in}

\noindent
Please remember when submitting any work via email or in person to\dots\\
\force \hfill \fbox{\fbox{\bf\large\color{red}PUT YOUR NAME ON YOUR WORK!}} \hfill \force

\vspace{0.2in}

\begin{description}
\item[\#1]  The Matrix\dots \qquad \qquad \dots{}problem

   \begin{enumerate}[(a)]
   \item Compute $A^{-1}B^2$ where $\displaystyle A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}, \quad B = \begin{bmatrix} 1 & 0 \\ 2 & 3 \end{bmatrix} \in \mbox{GL}_2(\mathbb{Z}_{13})$

   \vspace{0.05in}

{\it Hint:} Not so random fact $(-5)13+(6)11=1$. 

   \vspace{0.05in}

   \item Find the cyclic subgroup generated by $B = \begin{bmatrix} 1 & 0 \\ 2 & 3 \end{bmatrix}$ (in $\mathrm{GL}_2(\mathbb{Z}_{13})$). What is the order of $B$?

   \end{enumerate}

   \vspace{0.2in}
   
   \item[\#2] Orders of elements and number of such elements.
   \begin{enumerate}[(a)]
   \item Make a table which lists the possible orders of elements of $\mathbb{Z}_{825}$. List the number such elements in the second row.

\begin{tabular}{|r|c|c|c|} \hline
Order = & 1 & ??? & $\cdots$ \Large\strut\\ \hline
Number of such elements = & 1 & ??? & $\cdots$ \Large\strut\\ \hline
\end{tabular}

\vspace{0.1in}

           How many generators does $\mathbb{Z}_{825}$ have?

\vspace{0.1in}

{\it Suggestion:} Type ``divisors of 825'' into a tool like \href{https://www.wolframalpha.com/input?i=divisors+of+825}{WolframAlpha}.

\vspace{0.1in}
            
   \item Repeat part (a) for $D_{242}$ 

\begin{tabular}{|r|c|c|c|} \hline
Order = & 1 & ??? & $\cdots$ \Large\strut\\ \hline
Number of such elements = & 1 & ??? & $\cdots$ \Large\strut\\ \hline
\end{tabular}

\vspace{0.05in}

            Does $D_{242}$ have a generator? What is/are they? or Why not?

   \item How many elements of order 9 are there in $\mathbb{Z}_{12345654321}$? What is/are they? or Why are there none?

\vspace{0.05in}

   \item How many elements of order 6 are there in $\mathbb{Z}_{12345654321}$? What is/are they? or Why are there none?
\end{enumerate}

   \vspace{0.2in}

\item[\#3] Let $g \in G$ (for some group $G$). Suppose $|g|=132$. List the {\it distinct} elements of $\langle g^{110} \rangle$. 

   \vspace{0.2in}

\item[\#4] Let $x,y \in G$ (for some group $G$). If there exists some $g \in G$ such $gxg^{-1}=y$, we say $x$ and $y$ are {\it conjugates}.
   \begin{enumerate}[(a)]
   \item Use induction to show $(gxg^{-1})^k=gx^kg^{-1}$ for any $k \in \mathbb{Z}_{\geq 0}$. 

\vspace{0.05in}
   
   {\it Note:} Consequently, $(gxg^{-1})^n=e$ iff $gx^ng^{-1}=e$ iff $x^n=g^{-1}eg=e$, so conjugates have the same order.
   
\vspace{0.05in}

   \item Show $xy$ and $yx$ are conjugate.
      
   \item Putting (a) and (b) together, we have $|xy|=|yx|$ and so $\langle xy \rangle$ and $\langle yx \rangle$ have the same size.
   
   \vspace{0.05in}
   
   Prove or give a counterexample: $\langle xy \rangle = \langle yx \rangle$ (where $x,y \in G$). 
        
   \end{enumerate}
   
%   \vspace{0.2in}
%
%\item[RESUBMIT] Type up Test \#1 Problem(s) \#??? and its/their solution(s) in \LaTeX. 
%
%\fbox{\parbox{6in}{
%Quotation of problem(s).
%}}
%
%     When typing this problem(s) up, write it/them up carefully: Restate the problem(s). Write in complete sentences.  

\end{description}

\end{document}