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.