PGL(1,R) - Knowlpedia

$\operatorname{PGL}(1,\mathbb R)$ is the projective general linear Lie group

\operatorname{PGL}(1,\mathbb R)=\operatorname{GL}(1,\mathbb R)/\{\lambda I:\lambda\in \mathbb R^\times\},

with the quotient Lie group structure.

Dimensions and structure

This group has real dimension $0$ . It acts on projective space by applying a representative matrix to a line. In size one both numerator and scalar subgroup coincide (or are trivial), so the projective group is trivial.

References

Pavel Etingof, Lie Groups and Lie Algebras (MIT lecture notes), §6.1, pp. 38–39; Proposition 6.3 and Exercise 6.5.