\documentclass{article}
\usepackage{amssymb}
\usepackage{latexsym}
\title{Group Theory: Sheet 3}
\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 Do exercises 1.11, 1.12, 2.1 to 2.7 in
{\it Topics in Group Theory}. Only look at the answers at the 
back of the book if you get really stuck. 
\item Suppose that $\alpha: G \rightarrow H$ is an isomorphism of
groups. Show that the map $\alpha^{-1}: H \rightarrow G$ is
also an isomorphism.
\item 
\begin{enumerate}
\item[(a)] Suppose that $\alpha$ is an automorphism of a group $G$.
Let $F_\alpha = \{ g \in G \mid (g)\alpha = g\}$. Prove that
$F_\alpha \leq G$.
\item[(b)] Let $G = \mbox{GL}(n, \mathbb C)$ be the group of 
invertible $n$ by $n$ complex matrices. Consider the
map $\beta: G \rightarrow G$ which sends each matrix to 
the transpose of its inverse. Prove that $\beta$ is an 
automorphism of $G$. Deduce that the subset of $G$
consisting of those $g \in G$ such that $gg^T = I_n$ is a group.
\end{enumerate}
\item Determine all automorphisms of the additive group $\mathbb Z$.
Determine all homomorphisms $\gamma : \mathbb Z \rightarrow \mathbb Z$.
\item Determine all automorphisms of the additive group $\mathbb Q$
of rational numbers.
\item List all the automorphisms of $S_3$, and demonstrate 
that you have found them all.
\item Suppose that $H$ is a subgroup of a group $G$.
Let $\Omega = G/H$ so $\Omega$ is a collection of subsets
of $G$. Subsets of $G$ may be multiplied together.
Suppose that $\Omega$ is closed under multiplication. 
Prove that $\Omega$ is a group, and deduce that $H$ is a normal
subgroup of $G$.
\item Suppose that $G$ is a group with subgroups $H$ and $K$.
Suppose that both $H$ and $K$ have finite index in $G$.
Deduce that $H \cap K$ must have finite index in $G$.
\item Exhibit a group $G$ with the following property: the
intersection of all subgroups of finite index in $G$ is a 
subgroup of infinite index in $G$.
\end{enumerate}
\end{document}
 
