Projective outer linear group

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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.