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\noindent
\parbox{1.95in}{\bf Math 3110}
\hfill {\Large \bf Homework \#4} \hfill
\parbox{1.95in}{\bf \hfill Due: Fri., Feb. 23${}^\mathbf{rd}$, 2024}

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

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\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

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

Consider $\displaystyle A = \begin{bmatrix} 2 & 0 \\ 3 & 4 \end{bmatrix}$ and $\displaystyle B = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$ in $\mathbb{Z}_7^{2\times 2}$.



   \begin{enumerate}[(a)]
   \item Explain why these matrices belong to $\mbox{GL}_2(\mathbb{Z}_{7})$.
   \item Compute $A^2B^{-1}$.
   \item Find the cyclic subgroup generated by $A$ (working in $\mbox{GL}_2(\mathbb{Z}_{7})$). What is the order of $A$?

   \end{enumerate}

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   \item[\#2] Orders of elements and number of such elements.
   
   {\it Note:} It might be helpful to obtain a list of divisors of various numbers appearing here (like 350 and 98). I recommend using a tool like the website 
   \href{https://www.wolframalpha.com/}{WolframAlpha.com}. In Wolfram Alpha, you could type something like ``divisors of 350'' and it will give you a list.
   
   \begin{enumerate}[(a)]
   \item Make a table which lists the possible orders of elements of $\mathbb{Z}_{350}$. 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 $\mathbb{Z}_{350}$ have?
             
   \item Repeat part (a) for $D_{98}$ 

\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 $D_{98}$ have a generator? What is/are they? or Why not?

   \item How many elements of order 18 are there in $\mathbb{Z}_{12345678}$? What is/are they? or Why are there none?
   \item How many elements of order 18 are there in $\mathbb{Z}_{123456789}$? 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 $|g|=728$. List the {\it distinct} elements of $\langle g^{455} \rangle$. 

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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 Show the relation: $x \sim y$ iff $y$ is a conjugate of $x$ (i.e., there is some $g \in G$ such that $gxg^{-1}=y$) is an {\it equivalence relation}.
   
   In other words, show $\sim$ is reflexive: for all $x \in G$, we have $x \sim x$; \quad $\sim$ is symmetric: if $x \sim y$, then $y \sim x$; \quad and $\sim$ is transitive: if $x \sim y$ and $y \sim z$, then $x \sim z$.
   
   \item Since conjugacy is an equivalence relation, this partitions $G$ into equivalence classes: $$[x] = \{ y \in G \;|\; y \text{ is a conjugate of } x\}$$ 
   
   Suppose $G$ is abelian and $x \in G$. What is $[x]$?
   
   \item Find the conjugacy classes of $D_4 = \langle x,y \;|\; x^4=1, y^2=1, (xy)^2=1 \rangle = \{1,x,x^2,x^3,y,xy,x^2y,x^3y\}$. 
   \end{enumerate}
   
\end{description}

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