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\title{General Mathematics and CPS II}
\author{Exercise 9}
\date{March 2, 2012}
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\begin{enumerate}

\item Let $G$ be a group, and let $H$ and $K$ be subgroups of $G$.
Show that $H \cap K$ is also a subgroup of $G$.

\item Prove Ivanov p.\ 40, Lemma 4.  I.e., show that the
symmetry group of an ornament may contain, apart from the cyclic
subgroup of translations $\langle \Pi \rangle$, only the following:
\begin{enumerate}
\item A line reflection $R_\|$ about the axis of the ornament.
\item Line reflections about axes perpendicular to the ornament.
\item Point reflections about points on the axis of the ornament.
\item At most one \emph{glide reflection} about the axis of the
ornament, which is a composition of a translation and a line
reflection about the axis of the ornament.
\end{enumerate}

\item (Ivanov, p. 41, Exercise.)  Let $R_\alpha$ denote the reflection
about the line $x=\alpha$.  Let $G$ be the (symmetry) group generated
by the unit translation along the $x$-axis and by $R_0$.  Show that
$R_\alpha \in G$ if and only if $2 \alpha \in \mathbb{Z}$.

\end{enumerate}
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