Projective general linear group is almost simple

Statement
Let $$k$$ be a field and $$n$$ be a natural number greater than $$1$$. Then, if $$n \ge 3$$ or $$k$$ has at least four elements, the projective general linear group $$PGL_n(k)$$ is an almost simple group.

Facts used

 * 1) uses::Projective special linear group is centralizer-free in projective general linear group
 * 2) uses::Projective special linear group is simple