On 1/13/08, Fred lunnon <fred.lunnon@gmail.com> wrote:
... It's a shame this group is not better known --- I've several times seen claims, often by quite respectable mathematicians, to the effect that the "Moebius group is the largest group preserving spheres" --- which is quite simply untrue! The Lie sphere group is generated by the union of the Moebius and Laguerre groups. In n-space it has dimension (n+3)(n+2)/2, and (modulo various finite quotients about which I don't want to get into arguments here) the spin group is essentially O(2,n). I think of it as a third step in the sequence Euclidean group, Moebius group, Lie sphere group.
Department of Inevitable Corrections (DIC) --- that should have read O(n+1,2) --- indefinite orthogonal group. Notice that --- unlike Euclidean and spherical cases --- the full symmetry (Pin) group has in general 4 connected components, according to whether the associated (metric) quadratic form evaluates to positive or negative (timelike versus spacelike, in some order), as well as whether orientation (parity) is reversed. [I'm a little hazy about the precise matrix representation details --- I abandoned matrices for geometric symmetries a few years back, and now only use Geometric Algebra --- over to you, Dan.] WFL