Exercise 4, § 3

Let V be a finite dimensional real vector space, W a finite subgroup of $\mathbf{GL}(V)$ generated by reflections, $w$ an element of $W$, $V'$ a vector subspace of V stable under $w$, and $k$ the order of the restriction $w|V'$ of $w$ to $V'$. Show that there exists $x \in W$ of order $k$, leaving $V'$ stable, and such that $x|V' = w|V'$. (Let $W'$ be the set of elements of $W$ leaving fixed the points of $V'$. This group is generated by reflections, and $w$ permutes the chambers relative to $W'$; deduce that there exists $h \in W'$ such that $wh$ leaves stable a chamber of $W'$; show that we can take $x = wh$.)