\documentclass{article} \usepackage{amssymb} \usepackage{latexsym} \title{Group Theory: Sheet 2} \author{\copyright G. C. Smith 2001} \date{} \begin{document} \maketitle \textit{The copyright is waived for non-profit making educational purposes.} \begin{enumerate} \item Let $G = \mbox{Sym}(\{1,2,3,4\})$. Consider the subgroup $H = \{ \mbox{id}, (1,2,3), (1,3,2)\}.$ Explicity describe the sets $G/H$ and $H\backslash G$ (i.e. list the left and right cosets of $H$ in $G$). \item Suppose that $H \leq K \leq G$. Let $T$ be a left transversal for $H$ in $K$, and let $S$ be a left transversal for $K$ in $G$. \begin{enumerate} \item[(a)] Suppose that $t, t' \in T$ and $s, s' \in S$. Show that $st = s't'$ if and only if both $s = s'$ and $t = t'$. \item[(b)] Show that $ST$ is a left transversal for $H$ in $G$. \item[(c)] Deduce that $|G:H| = |G:K| \cdot |K:H|$. \end{enumerate} \item Suppose that $G$ is a group with a subgroup $H$. Suppose that $T$ is a left transversal for $H$ in $G$. Does it follow that $T$ must also be a right transversal for $H$ in $G$? Justify your answer. \item Consider two non-parallel lines in a Euclidean plane. Let one of the angles between these lines be $\theta$. Show that the composition of the reflections in this line is a rotation about the point of intersection. The rotation is through what angle? \item Let $H, K$ be subgroups of $G$. Suppose that $x, y \in G$ and $Hx = yK$. Does it follow that $H = K?$ Justify your answer. \end{enumerate} \end{document}