Oops, yeah they’re disconnected. (I don’t like orientation-reversing isometries.) Could amend to “the group contains such a group”, or “C admits such a group”. If we want to be really precise, C might also have to be stipulated to be nowhere constant.
This isn’t an important point but are you sure it’s connected? Seems to me like the mirror-reflections are disconnected from the pure-rotations.
Oops, yeah they’re disconnected. (I don’t like orientation-reversing isometries.) Could amend to “the group contains such a group”, or “C admits such a group”. If we want to be really precise, C might also have to be stipulated to be nowhere constant.