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\noindent
\parbox{2.25in}{\bf Math 3110-101}
\hfill {\Large \bf Homework \#4} \hfill
\parbox{2.25in}{\bf \hfill Due: Fri., Feb. $18^\mathrm{th}$, 2022}

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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

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

   \begin{enumerate}[(a)]
   \item Compute $A^{-1}B^2$ where $\displaystyle A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}, \quad B = \begin{bmatrix} 3 & 2 \\ 1 & 1 \end{bmatrix} \in \mbox{GL}_2(\mathbb{Z}_9)$

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   \item Find the cyclic subgroup generated by $A$. What is the order of $A$?

   \end{enumerate}

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   \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 $\mathbb{Z}_{495}$. List of 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}

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           How many generators does $\mathbb{Z}_{495}$ have?
             
   \item Repeat part (a) for $D_{44}$ 

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

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            Does $D_{44}$ have a generator? What is/are they? or Why not?

   \item How many elements of order 12 are there in $\mathbb{Z}_{29616}$? What is/are they? or Why are there none?
   \item How many elements of order 9 are there in $\mathbb{Z}_{29616}$? What is/are they? or Why are there none?
\end{enumerate}

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\item[\#3] Let $g \in G$ (for some group $G$). Suppose $|g|=264$. List the {\it distinct} elements of $\langle g^{220} \rangle$. 

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\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 Let $y=gxg^{-1}$ for some $g \in G$. Show that $|x|=|y|$ (i.e. conjugates have the same order).  
   \item Prove or give a counterexample: $\langle x \rangle = \langle gxg^{-1} \rangle$ (where $x,g \in G$). 
   
            In other words, is it true or not that conjugates generate the same cyclic subgroup?
   \end{enumerate}
   
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