Projective outer linear group

Definition
Suppose $$k$$ is a field and $$n$$ is a natural number. The projective outer linear group of degree $$n$$ over $$k$$, denoted $$POL(n,k)$$, is defined as the quotient group of the outer linear group $$OL(n,k)$$ by the center of the general linear group $$GL(n,k)$$ that sits as a normal subgroup in it.

Note that in case $$n = 1$$ and $$n = 2$$, this group is isomorphic to an external direct product of $$PGL(n,k)$$ and a cyclic group of order two, and is not interesting to study. For $$n \ge 3$$, this group is an external semidirect product of $$PGL(n,k)$$ and a cyclic group of order two and is not a direct product.