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On the Automorphisms of Infinite Chevalley Groups

Published online by Cambridge University Press:  20 November 2018

J. E. Humphreys
J. E. Humphreys*
Affiliation:
University of Oregon, Eugene, Oregon The Institute for Advanced Study, Princeton, New Jersey
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In (8, § 3.2) Steinberg proved the following result.

THEOREM. Let K be a finite field, G′ a simple Chevalley group (“normal type1’) over K. Then every automorphism of G’ is the composite of inner, graph, field, and diagonal automorphisms.

For the meaning of these notions, see (8). Our aim in this note is to indicate how the Theorem may be extended to arbitrary infinite fields K, provided we replace G′ by the group denoted G in (5) and ĝ in (8). This amounts to proving the Theorem for automorphisms of G′ which are induced by automorphisms of G; when K is finite, Steinberg's results show that all automorphisms of G′ arise in this way. As Steinberg points out, the sole use made of the finiteness of K in his argument is in the proof of the following statement: Let U be the subgroup of G′ corresponding to the set of positive roots, and let σ be any automorphism of G′; then Uσ is conjugate to U in G′.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 21 , 1969 , pp. 908 - 911
Copyright
Copyright © Canadian Mathematical Society 1969

References

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