What is the subgroup of the orthogonal group $O(n)$ generated by relections around $k$-dimensional hyperplanes?
For $k=n-1$ it is $O(n)$ by the Cartan's theorem. For $k=0$ it is $\{\pm Id \}$. What about the other cases?
Edit. By a reflection $S$ on $X$ I mean a linear orthogonal mapping such that $S^2=Id$. By a reflection around $k$-dimensional hyperplane $V\subset X$ I mean the reflection for which the set of fixed points (linear subspace) $V$ has dimension $k$.