\documentclass{article}
\usepackage{amsmath, amssymb,verbatim, amsthm, graphicx}

%exercise environment
\newcounter{exercise}
\setcounter{exercise}{0}
\newenvironment{exercise}{\addtocounter{exercise}{1} \noindent{\bf{Exercise \theexercise.}}}{\vspace{.5cm}}

\newcommand{\R}{\mathbb{R}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\C}{\mathbb{C}}

\textwidth 6.5in
\textheight 9in
\oddsidemargin 0in
\evensidemargin 0in
\headsep -0.5in

\setlength{\parskip}{0.25cm}

\pagenumbering{gobble}

\begin{document}

\begin{center}
\includegraphics[width=0.5in]{../ltr-logo.jpg}
\end{center}

\noindent{\Large{\textsc{Modern Algebra \hfill Assignment 2.2  \\ Spring 2019 \hfill Due January 30}}}

\large

\vspace{1cm}

\begin{exercise}
Let $G$ be a group and suppose that $x \in G$ has finite order $n$.  Prove that
\[
\{ m \in \Z \, | \, x^m=e  \} = n \Z,
\]
where $n \Z = \{ nk \, | \, k \in \Z  \}$.
\end{exercise}

\begin{exercise}
Let $G$ be a finite group.  Prove that if $G$ has even order, then $G$ contains an element with order 2.
\end{exercise}

\begin{exercise}
Let $G$ be a group and $a, b \in G$.  Denote the order of $a$ by $|a|$.  Prove the following assertions.
\begin{itemize}

\item[\bf a.]  $|a| = |a^{-1}|$

\item[\bf b.]  $|ab| = |ba|$

\item[\bf c.]  $|a|= |bab^{-1}|$

\end{itemize}
\end{exercise}

\end{document}
