We can all relax --- the universe is saved. For a while longer, at any rate. I can after all demonstrate Mob(n) as a subgroup of Lag(n+1) , using an unexpected geometrical twist in the construction. Starting with (say) Mob(2) acting on circles on the unit sphere in 3-space in the natural fashion, expand the circles to spheres orthogonal to the unit sphere --- these spheres are unoriented, and can be identified with Möbius inversions. Now apply the reflector (a properly Lie-sphere involution) interchanging Möbius inversion with Laguerre eversion (turning oriented spheres inside-out). The result is the subgroup of Lag(3) having reflector base planes meeting the origin; and adjoining Euc(3) translations yields Lag(3) as required. I wish I could show the proof of this --- it is still very neat (4 lines) --- with the algebra verified by computer (this time around). I might try putting up a crash-course explaining the background; however public reception of such attempts in the past has been less than ecstatic, to put it mildly ... It would be easy to overlook the extra involution in the absence of a common framework (the Lie-sphere group) in which to embed everything formally. I wonder if it is actually known to physicists, and if so what possible physical significance it might have? WFL << Unfortunately, another computer has pointed out that my "trivial proof" of Mob(n) + translations = Lag(n+1) is bunkum --- looks like my GA got a little rusty. Instead, it seems obvious that the natural embedding with Mob(2) acting on the unit sphere could not possibly work; from Adam's assertion that Poincaré(?) restricted to the unit sphere equals Möbius, it would follow that Poincaré does NOT equal Laguerre! So for me at any rate, it's back to square one on the relation between these groups when n = 2 . It might help if I had some clue what these people do with them once they're properly nailed down: eg. what group element corresponds to a change in velocity from rest to uniform along the x-axis? In the meantime, if you happen to notice the stars going out one by one tonight, you'll know what happened ...