Automorphisms of 𝐺𝐿_{𝑛}(𝑅)

Abstract

Let R denote a commutative ring having 2 a unit. Let G L n ( R ) {\text {G}}{{\text {L}}_n}\left ( R \right ) denote the general linear group of all n × n n\, \times \,n invertible matrices over R. Let ∧ \wedge be an automorphism of G L n ( R ) {\text {G}}{{\text {L}}_n}\left ( R \right ) . An automorphism ∧ \wedge is “stable” if it behaves properly relative to families of commuting involutions (see §IV). We show that if R is connected, i.e., 0 and 1 are only idempotents, then all automorphisms ∧ \wedge are stable. Further, if n ⩾ 3 n\, \geqslant \,3 , R is an arbitrary commutative ring with 2 a unit, and ∧ \wedge is a stable automorphism, then we obtain a description of ∧ \wedge as a composition of standard automorphisms.