\documentclass[12pt,letterpaper]{hmcpset}
\usepackage[margin=1in]{geometry}
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\name{}
\class{Math 171 - Abstract Algebra I}
\assignment{HW \#7}
\duedate{9/29/08}

\newcommand{\Z}{\mathbb{Z}}

\begin{document}

\problemlist{Dummit \& Foote (3.2) 16, 18, 19, 22}

\begin{problem}[3.2.16]
Use Lagrange's Theorem in the multiplicative group $(\Z/p\Z)^\times$ to prove \emph{Fermat's Little Theorem}: if $p$ is a prime then $a^p \equiv a \mod p$ for all $a \in \Z$.
\end{problem}
\newpage

\begin{problem}[3.2.18]
Let $G$ be a finite group, let $H$ be a subgroup of $G$ and let $N \unlhd G$.  Prove that if $|H|$ and $|G:N|$ are relatively prime then $H \leq N$.
\end{problem}
\newpage

\begin{problem}[3.2.19]
Prove that if $N$ is a normal subgroup of the finite group $G$ and $(|N|, |G:N) = 1$ then $N$ is the unique subgroup of $G$ of order $|N|$.
\end{problem}
\newpage

\begin{problem}[3.2.22]
Use Lagrange's Theorem in the multiplicative group $(\Z/n\Z)^\times$ to prove \emph{Euler's Theorem}:  $a^{\varphi(n)} \equiv 1 \mod n$ for every integer $a$ relatively prime to $n$, where $\varphi$ denotes Euler's $\varphi$-function.
\end{problem}

\end{document}