\documentclass[12pt]{amsart}
\usepackage{amsmath,amssymb,latexsym,enumitem}
\usepackage[letterpaper,margin=.9in]{geometry}

\begin{document}

\title{Math 120: Homework 5}
\author{}
\date{}
\maketitle

\begin{itemize}[noitemsep]
  \item Section 4.1 \# 1
  \item Section 4.2 \# 2,9
  \item Section 4.3 \# 4,22
  \item Section 7.3 \# 25,26 (a) and (c) only
\end{itemize}

\bigbreak\noindent
{\textbf{Problem 4.1 \#1.}} Let $G$ act on the set $X$. Prove that if $a, b
\in A$ with $b = g \cdot a$ for some $g \in G$ then $G_b = g G_a g^{- 1}$.
($G_a$ is the stabilizer of $a$.) Deduce that if $G$ acts transitively on $A$
then the kernel of the action is
\begin{equation}
  \label{kerggg} \bigcap_{g \in G} g G_a g^{- 1} .
\end{equation}

\bigbreak\noindent
{\textbf{Problem 4.2 \#2.}} List the elements of $S_3$ as $1, (12), (23),
(13), (123), (132)$ and label these with the integers $1, 2, 3, 4, 5, 6$
respectively. Exhibit the image of every element of $S_3$ under the left
regular representation of $S_3$ into $S_6$.

\bigbreak\noindent
{\textbf{Problem 4.2 \#9.}} Prove that if $p$ is a prime and $G$ is a group
of order $p^{\alpha}$ then every subgroup of $G$ of index $p$ is normal.
Deduce that every group of order $p^2$ has a normal subgroup of order $p$.

\bigbreak\noindent
{\textbf{Problem 4.3 \#4.}} Prove that if $S \subseteq G$ is a subset then
$g N_G (S) g^{- 1} = N_G (g S g^{- 1})$ and $g C_G (S) g^{- 1} = C_G (g S g^{-
1})$.

\medbreak\noindent
{\textbf{Problem 4.3 \#22.}} Show that if $n$ is odd, the set of all $n$-cycles in $A_n$ 
consists of two conjugacy classes of equal size.

\bigbreak\noindent
{\textbf{Problem 7.3 \# 25.}} Assume $R$ is a commutative ring with $1$.
Prove that the binomial theorem
\begin{equation}
  \label{bimo} (a + b)^n = \sum_{k = 0}^n \binom{n}{k} a^k b^{n - k}
\end{equation}
holds in $R$, where the binomial coefficient $\binom{n}{k}$ is interpreted as
$1 + 1 + \cdots + 1$ of the identity of $R$ taken $\binom{n}{k}$ times.

\bigbreak\noindent
{\textbf{Problem 7.3 \# 26.}} The {\textit{characteristic}} of a ring is the
smallest positive integer such that $1 + \cdots + 1 = 0$ ($n$ times) in $R$.
If no such positive integer exists then characteristic of $R$ is said to be
zero. For example $\mathbb{Z}/ n\mathbb{Z}$ is a ring of characteristic $n$
and $\mathbb{Z}$ is a ring of characteristic zero.

(a) Prove that the map $\mathbb{Z} \longrightarrow R$ defined by
\begin{equation}
  \label{charmap} k \longmapsto \left\{ \begin{array}{ll}
    \text{$1 + \cdots + 1$ \qquad ($k$ times)} \qquad & \text{if $k > 0$,}\\
    0 & \text{if $k = 0$},\\
    - (1 + \cdots + 1) \qquad \text{($- k$ times)} & \text{if $k < 0$}
  \end{array} \right.
\end{equation}
is a ring homomorphism whose kernel is $n\mathbb{Z}$ where $n$ is the
characteristic of $R$.

(c) Prove that if $p$ is a prime of $R$ and if $R$ is a commutative ring of
characteristic $p$ then $(a + b)^p = a^p + b^p$ for all $a, b \in R$.

\end{document}
\medbreak\noindent
