\documentclass[11pt]{article} %
\usepackage{fullpage}
\usepackage{graphicx}
\usepackage{graphics}
\usepackage{psfrag}
\usepackage{amsmath,amssymb}
\usepackage{enumerate}
\setlength{\textwidth}{6.5in}
\setlength{\textheight}{9in}
\newcommand{\cP}{\mathcal{P}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\Q}{\mathbb{Q}}
\begin{document}
\noindent
{\bf Math 307 Abstract Algebra \quad Homework 4}
\bigskip
Solve the following problems.
Five points for each problem.
\begin{enumerate}
\item Consider $\sigma = (13256)(23)(46512) \in S_5$.
(Read the first 2 pages of Chapter 5 for
the cycle notation.)
(a) Express $\sigma$ as a product of disjoint cycles.
(b) Express $\sigma$ as a product of transpositions.
(c) (Extra 2 points.) Express $\sigma$ as a product minimum number of
transpositions.
(Prove that the number is minimum!)
\item (a) Let $\alpha = (1,3,5,7,9,8,6)(2,4,10)$. Determine with proof the smallest
positive integer $n$
\quad such that
$\alpha^n = \alpha^{-5}$.
(b) Let $\beta = (1,3,5,7,9)(2,4,6)(8,10)$. If $\beta^m$ is a 5-cycle,
what can you say about $m$?
\item In $S_7$ show that $x^2 = (1,2,3,4)$ has no solutions, but $x^3 = (1,2,3,4)$
has at multiple solutions. (Extra 2 points if you determine all the solutions with proofs.)
Hint: If $x = \sigma_1 \cdot \sigma_k$ is a solution in disjoint cycle representation, then ...
\item Describe in terms of the disjoint cycle decomposition
all elements of order $5$ in $A_6$.
\item Let $H \le S_n$.
(a) Show that either $H \le A_n$ or $|H\cap A_n| = |H|/2$.
(b) If $|H|$ is odd, show that $H \le A_n$.
Hint: (a) If $H \cap A_n \ne H$, set up a bijection
from $H\cap A_n$ to $H - A_n$ in the latter case.
\item Let $G$ be a group. Show that $\phi: G \rightarrow G$ defined by
$\phi(g) = g^{-1}$ is an isomorphism if and only if $G$ is Abelian.
\item Recall that $U(n)$ is group containing
the elements in $Z_{n}$ that are relatively
prime to $n$ such that $a*b = ab$ (mod $n$).
Show that $\phi: U(16) \rightarrow U(16)$ defined by
$\phi(x) = x^3$ is an isomorphism (automorphism).
What about the maps $x \mapsto x^5$ and $x \mapsto x^7$?
\item Show that every isomorphism (automorphism) $\phi: (\Q,+) \rightarrow (\Q,+)$
has the form $\phi(x) = qx$ for $q = \phi(1)$.
\end{enumerate}
\end{document}