Correction: On Aug 31, 2014, at 10:20 AM, Dan Asimov <dasimov@earthlink.net> wrote:
Yes -- I think you used the word "preserve", which means exactly what you just said -- so I think what you said was clear.
Maybe we are using different terminology.
The next sentence should have said "(n-1)-spheres", not "circles":
Because with the terms I'm using, the group Conf(S^n) is generated by the inversions in circles (n-1)-spheres (okay, even numbers of inversions in circles (n-1)-spheres).
The orientation preserving projective group on S^n would be PGL+(n+1,R), which just permutes the lines through the origin in R^(n+1) and so induces a map of S^n to itself.
But (take S^2 for instance, where Conf(S^2) == PSL(2,C)) most elements of Conf(S^n) do not take antipodal pairs to antipodal pairs, whereas elements of PGL(n+1,R) always take antipodal pairs to antipodal pairs.
With this view, I don't understand:
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The [orientation-preserving] elements of the projective group PSL(n+1) acting on S^n are precisely the elements of the conformal group Conf(S^n).
--Dan