From: mareg@mimosa.csv.warwick.ac.uk ()
Subject: Re: Automorphis group of GL_n(K)
Date: 14 Feb 2001 08:58:08 GMT
Newsgroups: sci.math
Summary: Automorphisms of SL_n(K)

In article <96bq4g$1ccn$1@nef.ens.fr>,
	jriou@clipper.ens.fr (Joel Riou) writes:
>tim_brooks@my-deja.com, dans le message (sci.math:393299), a écrit :
>> Let K be a field and n>=2.
>> What is the automorphism group of GL_n(K) ?
>
>
>If you want to know the automorphisms of GL_n(K) as an abstract group,
>it might be strange because, for instance any automorphism of K as a
>field would provide an automorphism of GL_n(K). But, if you want to
>know the group of automorphism of GL_n(K) as an algebraic group
>defined over K then the answer is more interesting (as far as i am
>concerned).
>

[Description of the group of automorphisms of GL_n(K) as an algebraic group
 omitted.]

Sorry I missed the original posting. I sort of get the impression that there
is an unusaully high noise-signal ratio on sci.math at the moment. Has
anyone else noticed or am I just being over-sensitive?

Anyway, the automorphism group of SL_n(K) is < PGammaL_n(K), t>.

Here PGL_n(K) is the inner automorphism group, which is just GL_n(K)
modulo scalars, PGammaL_n(K) is PGL_n(K) extended by the group of
field automorphisms of K, and t is the "graph" automorphism, that
maps a matrix A to (A^T)^-1.

Aut(SL_n(K)) is contained in Aut(GL_n(K)), but GL_n(K) usually
has extra automorphismws that centralize SL_n(K), and have the form
A -> det(A)^i A, for some i>=0. These maps are all endomorphisms of
GL_n(K), but you have to be a bit careful, becasue they are not all
automorphisms.
For example |Aut(GL_2(7))| = 4|PGL(2,7)| and
|Aut(GL_2(11)| = 8|PGL(2,11)|.

Derek Holt.