nLab projective general linear group

Definition

The projective general linear group $PGL(n)$ (in some dimension $n$ and over some coefficients) is the quotient of the general linear group $GL(n)$ by its center, in other words by the subgroup consisting of scalar multiplies of the identity.

Elements of $PGL(n)$ are identified with the automorphisms of the projective space of lines in the original vector space where $GL(n)$ is acting, hence they amount to projective transformations of the projective space.

Examples

• $PGL(2)$ has a canonical action on the upper half plane and as such is equivalent to the modular group of Möbius transformations

Related concepts

• projective special linear group

• projective space

• projective unitary group

• special linear group

• Fuchsian group

• Kleinian group

• projective representation

• cross ratio

References

• Wikipedia, Projective linear group