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\begin{center}
{\large MATH 150A Winter 2020 \ -  \ {\bf Problem Set 9}}

% Put your name and section here.
{\large due March 13}
\end{center}


\begin{enumerate}

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	\item Let $m$ be an orientation-reversing isometry of $\RR^2$.  Prove algebraically that $m^2$ is a translation.

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	\item Find the conjugacy class of an isometry of $\RR^2$ of each of the following types.
	\begin{enumerate}
	 \item Translation.
	 \item Rotation about a point.
	 \item Reflection across a line.
	 \item Glide reflection across a line.
	\end{enumerate}
	
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	\item Let $\ell_1$ and $\ell_2$ be lines through the origin in $\RR^2$ that intersect at an angle of $\pi/n$ and let $r_i$ be the reflection across $\ell_i$.  Prove that $r_1$ and $r_2$ generate a dihedral group $D_n$.

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	\item Let $S$ and $S'$ be subsets of $\RR^n$.  $S$ is {\em dense} in $S'$ if for every point $a \in S'$ and every $\varepsilon > 0$, there is $s \in S$ with $|a - s| < \varepsilon$.
	\begin{enumerate}
	 \item Prove that an additive subgroup $G$ of $\RR$ is either dense in $\RR$ or else discrete.
	 \item Prove that the additive subgroup of $\RR$ generated by 1 and $\sqrt{2}$ is dense in $\RR$.
	 \item Let $H$ be a subgroup of $\SO_2$.  Prove that either $H$ is cyclic or dense in $\SO_2$.
	\end{enumerate}

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	\item Find the symmetry group of
	\begin{enumerate}
	 \item an I-beam, which one can think of as the product set of the letter I and an interval.
	 \item a baseball (or equivalently a tennis ball) accounting for the seam.
    \end{enumerate}

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\end{enumerate}
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