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

\usepackage{ulem}
\ULdepth 0 ex

\name{}
\class{Math 171 - Abstract Algebra I Section \uline{\;\;}}
\assignment{HW \#19}
\duedate{10/28/10}

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

\begin{document}

\begin{problem}
1. Let $ABC$ be an equilateral triangle, and let $D$ be its centroid. The group $G = D_{3}$ acts on the triangle in the way defined in class. Consider the action this gives on the set $X = \{ A,B,C,D\}$. 
\begin{itemize}
\item[(a.)] What are the orbits of the action of $G$ on $X$?
\item[(b.)] Find the subgroups $G_{A}$ and $G_{D}$.
\item[(c.)] What is $X_{G}$?
\end{itemize}
\end{problem}


\newpage

\begin{problem}[(Sometimes called the $n!$ theorem)] 

\noindent
2. Let $G$ be a finite group and let $H \leq G$ with $|G:H| = n$.
\begin{itemize}
\item[a.] Show that $G$ acts on the set $\{ xH \, | \, x \in G\}$ by left multiplication.
\item[b.] Show that $N$, the kernel of this action, is contained in $H$.
\item[c.] Show that $G/N$ is isomorphic to a subgroup of $S_{n}$.
\item[d.] Conclude that $|G:N|$ divides $n!$.
\end{itemize}
\end{problem}

\newpage

\begin{problem}
3. Let $G$ be a finite group and let $H \leq G$ with $|G:H| = p$, where $p$ is the smallest prime divisor of $|G|$. Apply the previous problem to show $H \lhd G$. (Compare this problem to question 5 from Exam 1.)

\end{problem}


\end{document}