In the complex plane it is known that given two disjoint circles we can find a Mobius transformation that maps them to concentric circles.
My question: Given two disjoint $(n-1)$-spheres $Y$ and $Z$ in $\mathbb{R}^n$ is there a Mobius transformation of $\mathbb{R}^n$ that maps them to concentric spheres or is that not true in general?
Without loss of generality, two circles are symmetric across the real axis, and so the Mobius transformation used to make them concentric is also symmetric across the real axis (see here).
This generalizes to spheres by symmetry: we can make spheres concentric using a Mobius transformation which is symmetric across the axis of symmetry (of the two spheres); just pick any plane containing this axis, get a 2D Mobius transformation, then extend to rest of space by symmetry.