Is every finite group of isometries in $d$-dimensional Euclidean space a subgroup of a finite group generated by reflections?
By "reflection" I mean reflection in a hyperplane: the isometry fixing a hyperplane and moving every other point along the orthogonal line joining it to the hyperplane to the same distance on the other side.
Every isometry is a product of reflections; in $d$-space, at most $d+1$ reflections are needed. So, every finite group of isometries is a subgroup of a group generated by reflections; just choose some reflections that yield each isometry, and take the group generated by those. The question is whether the reflections can always be chosen so that the resulting group is still finite, or even discrete.
Every finite group $G$ of isometries has a fixed point. If we regard this fixed point as the origin, then $G$ is a finite subgroup of the orthogonal group $O(d)$. Orthogonal transformations are the product of at most $d$ reflections in hyperplanes through the origin.
I have found some claims that the answer is "Yes", but it's not clear if they mean for arbitrary dimension, or just 3-space. Also, no proofs are provided. This monograph "Symmetry Groups" claims "All the finite groups are reflection groups or subgroups thereof" (p.11). This webpage says "All symmetry groups will be subgroups of groups generated by reflections."