\documentclass[12pt,letterpaper]{hmcpset}
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\name{}
\class{Math 171 - Abstract Algebra I}
\assignment{HW \#3}
\duedate{9/13/10}

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

\begin{document}

\problemlist{Dummit \& Foote (1.3) 10, 14. (1.4) 10. (1.5) 2}

\begin{problem}[1.3.10]
Prove that if $\sigma$ is the $m$-cycle $(a_1\ a_2\ \ldots\ a_m)$, then for all $i \in \{1,2,\ldots,m\}$, $\sigma^i(a_k) = a_{k+i}$, where $k+i$ is replaced by its least positive residue mod $m$.  Deduce that $|\sigma| = m$.
\end{problem}
\newpage

\begin{problem}[1.3.14]
Let $p$ be a prime.  Show that an element has order $p$ in $S_n$ if and only if its cycle decomposition is a product of commuting $p$-cycles.  Show by an explicit example that this need not be the case if $p$ is not prime.
\end{problem}
\newpage

\begin{problem}[1.4.10]
Let $G = \left\{ \begin{pmatrix} a & b \\ 0 & c \end{pmatrix} |\ a,b,c \in \R,\ a \neq 0,\ c \neq 0 \right\}.$
\begin{itemize}
\item[(a)] Compute the product of $\begin{pmatrix} a_1 & b_1 \\ 0 & c_1 \end{pmatrix}$ and $\begin{pmatrix} a_2 & b_2 \\ 0 & c_2 \end{pmatrix}$ to show that $G$ is closed under matrix multiplication.
\item[(b)] Find the matrix inverse of $\begin{pmatrix} a & b \\ 0 & c \end{pmatrix}$ and deduce that $G$ is closed under inverses.
\item[(c)] Deduce that $G$ is a subgroup of $GL_2(\R)$ (cf. Exercise 26, Section 1).
\item[(d)] Prove that the set of elements of $G$ whose two diagonal entries are equal (i.e., $a=c$) is also a subgroup of $GL_2(\R)$.
\end{itemize}
\end{problem}
\newpage

\begin{problem}[1.5.2]
Write out the group tables for $S_3$, $D_8$ and $Q_8$.
\end{problem}
\newpage

\end{document}