\documentclass{article} \usepackage{amssymb} \usepackage{latexsym} \title{Group Theory: Sheet 9} \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 Suppose that $\alpha: G \rightarrow H$ is a group epimorphism. Suppose that $K \leq G$ and let ${\overline K}$ denote the group $\{ (k)\alpha \mid k \in K\}$. Suppose that $L \leq H$. Let $\widehat L$ denote the group $\{ g \in G \mid (g)\alpha \in L\}.$ Let $N = \mbox{Ker }\alpha$. \begin{enumerate} \item Suppose that $A \leq G$. Show that $\widehat {\overline A} = AN$. \item Suppose that $B \leq H$. Show that $\overline {\widehat B} = B$. \item Suppose that $N \leq C \leq D \leq G$. Show that $|D:C| = |{\overline D}: {\overline C}|$ and that $C \unlhd D$ if and only if ${\overline C} \unlhd {\overline D}$. \item Suppose that $E \leq F \leq H$. Show that $|F:E| =|\widehat F: \widehat E|$. \item Show that bar'' and hat'' set up an inclusion preserving, index preserving bijection between the subgroups of $G$ which contain $N$, and the subgroups of $H$. \end{enumerate} \item Suppose that $G$ is a finite group and that $N \unlhd G$. Let $p$ be a prime number and suppose that $P \in \mbox{Syl}_p(N).$ Show that $G = N_G(P)N$. \item Suppose that $G$ is a finite group and that $p$ is a prime number. Let $P$ be a Sylow $p$ subgroup of $G$. \begin{enumerate} \item Suppose that $N_G(P) \leq H \leq G$. Prove that $N_G(H) = H$. \item Suppose that $N_G(P) \leq K \leq G$ and that $N_G(P) \leq L \leq G$. Furthermore suppose that there is $g \in G$ such that $H^g = K$. Prove that $H = K$. \end{enumerate} \end{enumerate} \end{document}