\documentclass{article}
\usepackage{amssymb}
\usepackage{latexsym}
\title{Group Theory: Sheet 6}
\author{\copyright G. C. Smith 2001}
\date{}
\begin{document}
\maketitle


\textit{The copyright is waived for non-profit making educational
purposes.}

{\it The course web site is available via 
{\tt http://www.bath.ac.uk/$\sim$masgcs/}}

\begin{enumerate}
\item Let $G$ be a group and let $G$ act on itself by conjugation.
Choose a transversal $\{ x_i \mid i \in I\}$ for the conjugacy classes.
\begin{enumerate}
\item[(a)] Suppose that $G$ is finite. 
Prove that \[|G| = \sum_{i \in I} |G:C_G(x_i)|\] by applying
the orbit-stabilizer theorem.
\item[(b)] Suppose that $G$ is finite of order $p^n$ where $n \geq 1$,
Use part (a) to show that $|Z(G)| \equiv 0 \mbox{ mod } p$
and deduce that $|Z(G)| > 1$.
\end{enumerate}
\item Let $G$ be a group of order $p^2$ where $p$ is a prime number.
\begin{enumerate}
\item[(a)] Show that if $G$ is non-abelian, we must have
$|Z(G)| = p$ and $g^p = 1$ for every $g \in G$.
\item[(b)] Suppose that $G$ is non-abelian, and choose $h \in G$
with $h \not \in Z(G)$. Let $H = \langle h \rangle.$ Show that
$H \leq N_G(H)$ and $Z(G) \leq N_G(H)$, and deduce that
$H \unlhd G$.
\item[(c)] Prove that $Z(G)H = G$ and $Z(G) \cap H = 1$.
Deduce that every group of order $p^2$ is abelian.
\end{enumerate}
\item Let $G$ be a finite group of even order. Show that the
number of elements which do not have order 2 is even, and 
deduce that $G$ contains an element of order $2$.
\item Let $G$ be a finite group and suppose that $p$ is a prime
number which divides $|G|$. Let 
\[ \Omega = \{ (g_1, g_2, \ldots, g_p) \mid g_1 g_2 \ldots g_p = 1\}.\]
\begin{enumerate}
\item[(a)] Prove that $|\Omega| = |G|^{p-1}$.
\item[(b)] Let $H = \langle x \rangle$ be a cyclic group of
order $p$. Define a multiplication by 
$(g_1, g_2, \ldots, g_p) \cdot x^i =
(g_{1+i}, g_{2+i}, \ldots, g_{p+i})$ for each
$(g_1, g_2, \ldots, g_p) \in \Omega.$ Show that every product 
$(g_1, g_2, \ldots, g_p) \cdot x^i$ is in $\Omega$
and demonstrate that this multiplication gives an 
action of $H$ on $\Omega$.
\item[(c)] Show that each orbit of $H$ acting on $\Omega$
has size $1$ or $p$.
\item[(d)] Describe the orbits of $H$ on $\Omega$ which have size 1.
\item[(f)] Use the orbit-stabilizer theorem to prove Cauchy's result
that if a prime number $p$ divides the order of a finite group,
then there is $g \in G$ with $g$ of order $p$ (i.e. $|\langle g \rangle|
=p$).
\item[(g)] How does Question 4(f) relate to Question 3?
\end{enumerate}
\item Suppose that $H \leq G$. Show that the number of conjugates
subgroups of the form $H^g$ (as $g$ ranges over $G$) is a divisor
of $|G|$.
\item Suppose that $G$ is a group and that $H$ is a subgroup
of $G$. If $g \in N_G(H)$, show that $\tau_g : H \rightarrow H$
defined by $h \mapsto h^g$ for every $h \in H$ is an
automorphism of $H$. Show that $\tau: g \mapsto \tau_g$
is a homomorphism from $N_G(H)$ to $\mbox{Aut }(H)$.
Deduce that $C_G(H) \unlhd N_G(H)$ and that
$N_G(H)/C_G(H)$ is isomorphic to a subgroup of 
$\mbox{Aut }(H)$.
\end{enumerate}
\end{document}
