\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}