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

\input xy
\xyoption{all}

%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 8.2  \\ Spring 2023 \hfill Due March 8}}}

\large

\vspace{1cm}

\begin{exercise}
Let $f:G \to H$ be a group homomorphism.  Use the First Isomorphism Theorem to show that
\[
[G: \ker f] = |\operatorname{im} f|.
\]
\end{exercise}

\begin{exercise}
Let $A$ be an additive abelian group.  Given $n \in \N$, recall that
\[
A[n] = \{ a \in A \,| \, na = 0 \}
\]
is the {\em $n$-torsion subgroup} of $A$.  Use the endomorphism $A \to A$ given by $a \mapsto na$ and the First Isomorphism Theorem to show that $A/A[n]$ is isomorphic to a subgroup of $A$. 
\end{exercise}

\begin{exercise}
If $G$ is a group and $a, b \in G$, the {\em commutator} of $a$ and $b$ is
\[
[a,b] = aba^{-1} b^{-1}.
\]
The {\em commutator subgroup} of $G$ is the subgroup generated by the set of commutators in $G$:
\[
[G,G] = \langle \, [a,b] \, | \, a,b \in G \rangle.
\]
Use the First Isomorphism Theorem to show that if $f :G \to H$ is a homomorphism and $H$ is abelian, then $[G,G] < \ker f$.
\end{exercise}

\end{document}
