I'm looking at a group of transformations from $X \cup \lbrace \infty \rbrace$ to itself, where $X$ is a Hilbert Space, and ultimately I'm looking for an appropriate name for this group.
The elements of group, as it turns out to be the products of isometries and non-zero scaling operators on $X$ (where $\infty$ maps to itself) and inversions in spheres. As such, these maps locally preserve angles, but not necessarily orientation. This makes them isogonal maps. Moreover, they are also smooth (where appropriate).
I thought about calling this group the "isogonal group", but this suggests that it's the group of isogonal maps, and I'm not sure the two concepts perfectly line up. I have heard of Liouville's Theorem of Conformal Mappings, but given this is possibly infinite-dimensional and as far as I can tell, the techniques used for proving this theorem do use finite dimensions (things like Jacobeans), I'd prefer a different name.
Surely, on $\mathbb{C}$, there's a name already for the group of Mobius Transforms and their conjugates? If such a name already exists, I think this would be the perfect name for this group.