\documentclass[12pt,letterpaper]{hmcpset}
\usepackage[margin=1in]{geometry}
\pagestyle{empty}

\name{}
\class{Math 171 - Abstract Algebra I}
\assignment{HW \#8}
\duedate{9/30/10}

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

\begin{document}

\problemlist{Dummit \& Foote (3.3) 3, 7, 8, 9}

\begin{problem}[3.3.3]
Prove that if $H$ is a normal subgroup of $G$ of prime index $p$ then for all $K \leq G$ either
\begin{itemize}
\item[(i)] $K \leq H$ or
\item[(ii)] $G=HK$ and $|K : K \cap H| = p$.
\end{itemize}

\end{problem}
\newpage

\begin{problem}[3.3.7]
Let $M$ and $N$ be normal subgroups of $G$ such that $G=MN$.  Prove that \\
$G/(M \cap N) \cong (G/M) \times (G/N)$. [Draw the lattice.]
\end{problem}
\newpage

\begin{problem}[3.3.8]
Let $p$ be a prime and let $G$ be the group of $p$-power roots of 1 in $\C$ (cf. Exercise 18, section 2.4).  Prove that the map $z \mapsto z^p$ is a surjective homomorphism.  Deduce that $G$ is isomorphic to a proper quotient of itself.
\end{problem}
\newpage

\begin{problem}[3.3.9]
Let $p$ be a prime and let $G$ be a group of order $p^am$, where $p$ does not divide $m$.  Assume $P$ is a subgroup of $G$ of order $p^a$ and $N$ is a normal subgroup of $G$ of order $p^bn$, where $p$ does not divide $n$.  Prove that $|P \cap N| = p^b$ and $|PN/N| = p^{a-b}$.  (The subgroup $P$ of $G$ is called a \emph{Sylow} $p$\emph{-subgroup} of $G$.  This exercise shows that the intersection of any Sylow $p$-subgroup of $G$ with a normal subgroup $N$ is a Sylow $p$-subgroup of $N$.)
\end{problem}

\end{document}