Projective Line - Symmetry Group

Symmetry Group

Quite generally, the group of Möbius transformations with coefficients in K acts on the projective line P1(K). This group action is transitive, so that P1(K) is a homogeneous space for the group, often written PGL₂(K) to emphasise its definition as a projective linear group. Transitivity says that any point Q may be transformed to any other point R by a Möbius transformation. The point at infinity on P1(K) is therefore an artifact of choice of coordinates: homogeneous coordinates

=

express a one-dimensional subspace by a single non-zero point (X,Y) lying in it, but the symmetries of the projective line can move the point ∞ = to any other, and it is in no way distinguished.

Much more is true, in that some transformation can take any given distinct points Q_i for i = 1,2,3 to any other 3-tuple R_i of distinct points (triple transitivity). This amount of specification 'uses up' the three dimensions of PGL₂(K); in other words, the group action is sharply 3-transitive. The computational aspect of this is the cross-ratio. Indeed, a generalized converse is true: a sharply 3-transitive group action is always (isomorphic to) a generalized form of a PGL₂(K) action on a projective line, replacing "field" by "KT-field" (generalizing the inverse to a weaker kind of involution), and "PGL" by a corresponding generalization of projective linear maps.