Automorphism group of general linear group over a field

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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.