# 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.