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

\usepackage{wasysym}

\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 3110-101}
\hfill {\Large \bf Homework \#4} \hfill
\parbox{2.25in}{\bf \hfill Due: Wed., Sept. $30^\mathrm{th}$, 2020}

\vspace{0.2in}

\begin{enumerate}
\item Orders of elements and number of such elements.
   \begin{enumerate}
   \item Make a table which lists the possible orders of elements of $\mathbb{Z}_{294}$. List of the number such elements in the second row.
             [I'll get you started: There is 1 element of order 1 \smiley] \quad  \underline{How many generators does $\mathbb{Z}_{294}$ have?}
   \item Repeat part (a) for $D_{294}$ \quad [Does $D_{294}$ have a generator? What is/are they or why not?]
   \item How many elements of order 8 are there in $\mathbb{Z}_{1440000}$? What is/are they? or Why are there none?
   \item How many elements of order 7 are there in $\mathbb{Z}_{1440000}$? What is/are they? or Why are there none?
\end{enumerate}

\item Let $g \in G$ (for some group $G$). Suppose $|g|=120$. List the {\it distinct} elements of $\langle g^{100} \rangle$. Is $g^{30} \in \langle g^{100} \rangle$?

\item Let $g,x \in G$ (for some group $G$). 
   \begin{enumerate}[i.]
   \item Show that $|x|=|gxg^{-1}|$ (i.e. conjugates have the same order).  
   \item Prove or give a counterexample: $\langle x \rangle = \langle gxg^{-1} \rangle$.
   \end{enumerate}
   
\item For each of the following permutations: 
   \begin{enumerate}[i.]
   \item Write the permutation as a product of disjoint cycles.
   \item Find its inverse.
   \item Find its order.
   \item Write it as a product of transpositions and state whether it is even or odd.
   \item Conjugate it by $\sigma = (123)(45)$ (i.e. compute $\sigma \tau \sigma^{-1}$).
   \item Compute $\tau^{99}$.
   \end{enumerate}

    \begin{enumerate}
    \item $\tau = (124)(35)(24)(132)$  
    \item $\tau = (1253)(354)(135)$  
    \item $\tau = (12435)(134)(45)$
    \end{enumerate} 

\item Orders in $S_n$.
   \begin{enumerate}
   \item What are the orders of the elements in $S_5$? Give an example of an element with each order.
   \item Does $S_{11}$  have an element of order 24? If so, find one. If not, explain why not.
   \item Does $S_{11}$  have an element of order 16? If so, find one. If not, explain why not. 
   \end{enumerate}

  
\end{enumerate}



\end{document}