Automorphism group of general linear group over a field

Definition

Let $k$ be a field and $n$ be a natural number. The group we are interested in is the automorphism group of the general linear group $GL(n,k)$ of degree $n$ over $k$. The automorphism group is denoted $\operatorname{Aut}(GL(n,k))$.

Structure

The structure of the automorphism group is as follows. It has a split short exact sequence:

$1 \to \operatorname{CAut}(GL(n,k)) \to \operatorname{Aut}(GL(n,k)) \to \operatorname{Aut}(PGL(n,k)) \to 1$.

Here, $\operatorname{CAut}(GL(n,k))$ is the central automorphism group of general linear group, which is given by automorphisms of the form:

$A \mapsto A \varphi(\det A)$

where $\varphi$ is a homomorphism from the multiplicative group of $k$ to itself chosen such that $x \mapsto x \varphi(x^n)$ is an automorphism of the multiplicative group of $k$.

$\operatorname{Aut}(PGL(n,k))$ is the automorphism group of projective general linear group over a field, i.e., the automorphisms of the quotient of $GL(n,k)$ by its center. This in turn can be expressed as:

$PGL(n,k) \rtimes (\operatorname{Aut}(k) \times C_2)$

where $\operatorname{Aut}(k)$ is the group of field automorphisms of $k$ with a natural induced action on $PGL(n,k)$, and $C_2$ is the cyclic group of order two acting via the transpose-inverse map.