Projective Linear Group - Name

Name

The name comes from projective geometry, where the projective group acting on homogeneous coordinates (x₀:x₁: … :x_n) is the underlying group of the geometry. Stated differently, the natural action of GL(V) on V descends to an action of PGL(V) on the projective space P(V).

The projective linear groups therefore generalise the case PGL(2,C) of Möbius transformations (sometimes called the Möbius group), which acts on the projective line.

Note that unlike the general linear group, which is generally defined axiomatically as "invertible functions preserving the linear (vector space) structure", the projective linear group is defined constructively, as a quotient of the general linear group of the associated vector space, rather than axiomatically as "invertible functions preserving the projective linear structure". This is reflected in the notation: PGL(n,F) is the group associated to GL(n,F), and is the projective linear group of (n − 1)-dimensional projective space, not n-dimensional projective space.