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\title{Group Theory: Sheet 2}
\author{\copyright G. C. Smith 2001}
\date{}
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\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}
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