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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Automorphisms of $ {\rm GL}\sb{n}(R)$


Author: B. R. McDonald
Journal: Trans. Amer. Math. Soc. 246 (1978), 155-171
MSC: Primary 20G99
MathSciNet review: 515534
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Abstract: Let R denote a commutative ring having 2 a unit. Let $ {\text{G}}{{\text{L}}_n}\left( R \right)$ denote the general linear group of all $ n\, \times \,n$ invertible matrices over R. Let $ \wedge $ be an automorphism of $ {\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\, \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.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1978-0515534-1
PII: S 0002-9947(1978)0515534-1
Keywords: Automorphisms, general linear group, commutative ring
Article copyright: © Copyright 1978 American Mathematical Society