\documentclass[12pt]{article}
\usepackage{fullpage}
\usepackage{amsfonts}
\usepackage{amsmath, amsthm, amssymb}
\pagestyle{headings}

\newcommand{\Q}{{\mathbb Q}}
\newcommand{\Z}{{\mathbb Z}}
\newcommand{\normal}{\trianglelefteq}
\begin{document}
\thispagestyle{empty}

\begin{center}
\textbf{Math 71}\\ Homework Assignment 20 October 1999
\end{center}



\begin{itemize}
\item p. 90:  41
\item p. 102:  3 [Hint:  Use the Second Isomorphism Theorem], and do
  not assume any of the groups are finite.
\item New Proof of Second Isomorphism Theorem (Theorem 18, p. 98):
Let $G$ be a group with $A, B$ subgroups of $G$ and with $B
\normal G$. Then $A \cap B \normal A$ and $AB/B \cong
A/A\cap B$.

Be sure to verify that $AB$ is a subgroup of $G$, $B \normal
AB$, and $A \cap B \normal A$.

Then proceed with a proof by justifying that $\varphi: A \to AB/B$
induced by a natural composition of maps $A \to AB \to AB/B$
($ a \mapsto a\cdot 1 \mapsto aB$) is a surjective homomorphism.
Computing the kernel of $\varphi$ and applying the first isomorphism
theorem should yield the result.
\end{itemize}


\end{document}