Exercise 2, § 3

Let $V$ be a finite dimensional real vector space equipped with a scalar product, $F$ a finite subgroup of the orthogonal group of $V$ generated by reflections, $\Lambda$ a discrete subgroup of $V$ stable under $F$, and $W$ the group of displacements of $V$ generated by $F$ and the translations by a vector belonging to $\Lambda$. Let $\mathfrak{H}$ be the set of hyperplanes $H$ of $V$ such that $s_H \in F$. Let $R$ be the set of elements of $\Lambda$ orthogonal to an element of $\mathfrak{H}$.

a) $W$ is generated by reflections if and only if $R$ generates $\Lambda$.

b) In $\mathbf{R}^2$ equipped with the scalar product $((x, y), (x', y')) \mapsto xx' + yy'$, let

$F$ the dihedral group generated by the $s_{\Delta_i}$, and $\Lambda$ the discrete subgroup of $\mathbf{R}^2$ generated by the $e_i$, a subgroup that is stable under $F$. Show that $W$ is not generated by reflections.