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 �crit : >> 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.