Automorphism group of general linear group over a field
The structure of the automorphism group is as follows. It has a split short exact sequence:
Here, is the central automorphism group of general linear group, which is given by automorphisms of the form:
where is a homomorphism from the multiplicative group of to itself chosen such that is an automorphism of the multiplicative group of .
is the automorphism group of projective general linear group over a field, i.e., the automorphisms of the quotient of by its center. This in turn can be expressed as:
where is the group of field automorphisms of with a natural induced action on , and is the cyclic group of order two acting via the transpose-inverse map.