Projective outer linear group
Suppose is a field and is a natural number. The projective outer linear group of degree over , denoted , is defined as the quotient group of the outer linear group by the center of the general linear group that sits as a normal subgroup in it.
Note that in case and , this group is isomorphic to an external direct product of and a cyclic group of order two, and is not interesting to study. For , this group is an external semidirect product of and a cyclic group of order two and is not a direct product.