\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}}
\newcommand{\mat}[4]{\left( \begin{array}{cc} #1 & #2 \\ #3 & #4  \end{array} \right)}
\newcommand{\smat}[4]{\left( \begin{smallmatrix} #1 & #2 \\ #3 & #4  \end{smallmatrix} \right)}

\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 11.2  \\ Spring 2019 \hfill Due April 17}}}

\large

\vspace{1cm}


\begin{exercise}
Let $C$ be an (additive) cyclic group of order $n$, generated by $a$, and suppose $n = de$ for some $d, e \in \N$.  Prove that
\[
C_d = \left\langle e a \right\rangle.
\]
\end{exercise}

\begin{exercise}  
Let $p$ be a prime and let $G$ be a group.  We say that $G$ is a {\em $p$-group} if the order of any element of $G$ is a power of $p$.\footnote{If $G$ is finite and abelian, we have proven that this definition is equivalent to $|G|$ being a power of $p$.  If we assume only that $G$ is finite, the same conclusion holds, but requires a different proof.  }  Prove the following statements about a $p$-group $G$.

\begin{itemize}

\item[\bf a.]  Let $x, y \in G$.  If $m = \max\{|x|,|y| \}$, then $x^m = y^m = e$.

\item[\bf b.]  If $m = \max \{|x| \, : \, x \in G\}$ is finite, then $m$ is an exponent for $G$.

\end{itemize}

\end{exercise}

\begin{exercise}
Given a prime $p$, an {\em elementary abelian $p$-group} is an abelian group in which every nonidentity element has order $p$.\footnote{Under modular arithmetic (addition {\em and} multiplication), the factor group $\Z/p\Z$ is a {\em field}, and the elementary abelian $p$-groups are precisely the $\Z/p\Z$-vector spaces.}

\begin{itemize}

\item[\bf a.]  Use the fundamental theorem of finite abelian $p$-groups (as we called it) to give a classification of all finite elementary abelian $p$-groups.

\item[\bf b.]  Can you think of a countably infinite elementary abelian $p$-group?  How about uncountable?

\end{itemize}

\end{exercise}


\end{document}
