Well, yes. The projective group PGL(n+1) acting on P^n or S^n has dimension dim(PGL(n+1)) = n^2 + 2n and the conformal group acting on S^n has dimension dim(Conf(S^n)) = (n+1)(n+2)/2 = (1/2)n^2 + (3/2)n + 1 -- just as Fred said. And, any element g of the projective group on S^n satisfies g(-p) = -g(p) for any point p of S^n, whereas that is not true for most elements of the conformal group on S^n. In fact, I wonder if the elements of the conformal group Conf(S^n) that happen to also be elements of [the projective group PLG(n+1) acting on S^n] are just the rotations. (This is certainly true for n = 2, where Conf(S^2) = Aut(S^2) = PSL(2,C), the holomorphic automorphism group of S^2. --Dan On Aug 29, 2014, at 5:53 PM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
Dan wrote << I've been suspecting that in fact the conformal and projective groups on S^n, as I've described them, are identical. Have not proved this yet. But if it's true, it must be well-known.
But (qua topological groups) their dimensions are respectively (n+2)(n+1)/2 versus (n+1)^2 - 1 ?!
Fred Lunnon