# Projective outer linear group

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.