Automorphism group of general linear group over a field
Definition
Let be a field and
be a natural number. The group we are interested in is the automorphism group of the general linear group
of degree
over
. The automorphism group is denoted
.
Structure
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.