\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 7.2  \\ Spring 2019 \hfill Due March 6}}}

\large

\vspace{1cm}

\begin{exercise}
Prove that $\Q$ has no proper subgroups of finite index.
\end{exercise}

\begin{exercise}
Prove that $\Q/\Z$ has elements of every finite order.
\end{exercise}

\begin{exercise}
Let $G$ be a group, $S \subset G$ and $H = \langle S \rangle$.  Fix $x \in G$ and set $K = \langle xSx^{-1} \rangle$.  

\begin{itemize}

\item[\bf a.] Prove that $K = xHx^{-1}$.  [{\em Suggestion:}  Prove that the subgroups of $G$ containing $xSx^{-1}$ are precisely the $x$-conjugates of the subgroups containing $S$.]

\item[\bf b.]  Conclude that if $xSx^{-1} \subset S$ for all $x \in G$, then $H \lhd G$.

\end{itemize}
\end{exercise}

\begin{exercise}
Given a group $G$, its {\em commutator subgroup} is defined to be
\[
G' = \langle xyx^{-1}y^{-1} \, | \, x, y \in G \rangle.
\]

\begin{itemize}

\item[\bf a.]  Prove that $G' \lhd G$.  [{\em Suggestion:}  Use the preceding exercise.]

\item[\bf b.]  If $H \lhd G$, prove that $G/H$ is abelian if and only if $G' < H$.  Thus $G/G'$ is the largest abelian factor group of $G$.

\item[\bf c.]  If $G' < H < G$, prove that $H \lhd G$.   

\end{itemize}

\end{exercise}

\end{document}
