Projective general linear group

Definition

Let $V$ be a nonzero finite-dimensional vector space over a field $k$ . The projective general linear group is the quotient group

\operatorname{PGL}(V) :=\operatorname{GL}(V)/(k^\times I),

where $k^\times I=\{\lambda I_V:\lambda\in k^\times\}$ is the central subgroup of nonzero scalar maps. For $V=k^n$ , it is denoted $\operatorname{PGL}_n(k)$ .

Action on projective space

The action of $\operatorname{GL}(V)$ on $\mathbb P(V)$ has kernel $k^\times I$ , so it descends to a faithful action of $\operatorname{PGL}(V)$ . Its elements are exactly the projective transformations induced by invertible linear maps.

For $\dim_kV\ge2$ , this action is sharply determined by its effect on a projective frame. It is generally smaller than the full collineation group when $k$ has nontrivial field automorphisms; the latter is the projective semilinear group.

Relation to

\operatorname{PSL}

The image of $\operatorname{SL}_n(k)$ in $\operatorname{PGL}_n(k)$ is the projective special linear group. It need not be all of $\operatorname{PGL}_n(k)$ . The obstruction is measured by determinant modulo $n$ -th powers in the $\operatorname{PGL}$ – $\operatorname{PSL}$ comparison.

References

James S. Milne, Algebraic Groups: The Theory of Group Schemes of Finite Type over a Field, Cambridge University Press, 2017. Author-maintained text. Relevant: classical algebraic groups and central quotients.
Emil Artin, Geometric Algebra, Interscience, 1957. Relevant: Chapter II, projective transformations.