\documentclass[12pt,letterpaper]{hmcpset}
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\name{}
\class{Math 171 - Abstract Algebra I}
\assignment{HW \#5}
\duedate{9/20/10}

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

\begin{document}

\problemlist{Dummit \& Foote (3.1) 1, 6, 7, 9}

\begin{problem}[3.1.1]
Let $\varphi:G \rightarrow H$ be a homomorphism and let $E$ be a subgroup of $H$.  Prove that $\varphi^{-1}(E) \leq G$ (i.e., the preimage or pullback of a subgroup under a homomorphism is a subgroup).  If $E \trianglelefteq H$ prove that $\varphi^{-1}(E) \trianglelefteq G$.  Deduce that $\ker \varphi \trianglelefteq G$.
\end{problem}
\newpage

\begin{problem}[3.1.6]
Define $\varphi: \R^\times \rightarrow \{ \pm  1\}$ by letting $\varphi(x)$ be $x$ divided by the absolute value of $x$.  Describe the fibers of $\varphi$ and prove that $\varphi$ is a homomorphism.
\end{problem}
\newpage

\begin{problem}[3.1.7]
Define $\pi : \R^2 \rightarrow \R$ by $\pi((x,y)) = x+y$.  Prove that $\pi$ is a surjective homomorphism and describe the kernel and fibers of $\pi$ geometrically.
\end{problem}
\newpage

\begin{problem}[3.1.9]
Define $\varphi : \C^\times \rightarrow \R^\times$ by $\varphi(a+bi) = a^2 + b^2$.  Prove that $\varphi$ is a homomorphism and find the image of $\varphi$.  Describe the kernel and the fibers of $\varphi$ geometrically (as subsets of the plane).
\end{problem}

\end{document}