im trying to show that the isometry of $\mathbb{R}^3$ given by $$ F(x,y,z) = (-y,x,z+1) $$ can be written as a composition of $4$ reflections and no less.
I know it can be written with at most $4$ reflections. I know it can't be written using $1$ or $3$ reflections, as that would reverse the orientation. But I'm not sure how I would go about showing that the isometry can't be written as a product of $2$ reflections.
We are left to show, that it cannot be a product of two reflections. Assume it is. The two reflection planes have nonempty intersection, otherwise they are parallel and in this case it would be a translation, which it isn't. The point in this nonempty intersection is a fixed point. But your map doesn't have any fixed points (look at the third entry).