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

\name{}
\class{Math 171 - Abstract Algebra I}
\assignment{HW \#4}
\duedate{9/16/10}

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

\begin{document}

\problemlist{Dummit \& Foote 1.6 \# 7, 13, 14, 23}

\begin{problem}[1.6.7]
Prove that $D_8$ and $Q_8$ are not isomorphic.
\end{problem}
\newpage

\begin{problem}[1.6.13]
Let $G$ and $H$ be groups and let $\varphi:G \rightarrow H$ be a homomorphism.  Prove that the image of $\varphi$, $\varphi(G)$, is a subgroup of $H$ (cf. Exercise 26 of Section 1).  Prove that if $\varphi$ is injective then $G \cong \varphi(G)$.
\end{problem}
\newpage

\begin{problem}[1.6.14]
Let $G$ and $H$ be groups and let $\varphi:G \rightarrow H$ be a homomorphism.  Define the \emph{kernel} of $\varphi$ to be $\{g \in G\ |\ \varphi(g) = 1_H\}$ (so the kernel is the set of elements in $G$ which map to the identity of $H$, i.e., is the fiber over the identity of $H$).  Prove that the kernel of $\varphi$ is a subgroup (cf. Exercise 26 of Section 1) of $G$.  Prove that $\varphi$ is injective if and only if the kernel of $\varphi$ is the identity subgroup of $G$.
\end{problem}
\newpage

\begin{problem}[1.6.23]
Let $G$ be a finite group which possesses an automorphism $\sigma$ (cf. Exercise 20) such that $\sigma(g) = g$ if and only if $g=1$.  if $\sigma^2$ is the identity map from $G$ to $G$, prove that $G$ is abelian (such an automorphism $\sigma$ is called \emph{fixed point free} of order 2). [Show that every element of $G$ can be written in the form $x^{-1}\sigma(x)$ and apply $\sigma$ to such an expression.]

\end{problem}
\newpage

\end{document}