Hostname: page-component-848d4c4894-8kt4b Total loading time: 0 Render date: 2024-06-27T12:49:54.506Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

1. Borel, A., Groupes linéaires algébriques, Ann. of Math. (2) 64 (1956), 20–82.Google Scholar
2. Borel, A. and Tits, J., Groupes réductifs, Publ. Math. I.H.E.S. no. 27 (1965), 55–150.Google Scholar
3. Borel, A. and Tits, J., Abstract homomorphisms of algebraic groups (to appear).Google Scholar
4. Chevalley, C., Théorie des groupes de Lie, volumes II et III (Hermann, Paris, 1951, 1955).Google Scholar
5. Chevalley, C., Sur certains groupes simples, Tôhoku Math. J. 7 (1955), 14–66.Google Scholar
6. Chevalley, C., Classification des groupes de Lie algébriques; Séminaire C. Chevalley, 1956-1958 (Secrétariat mathématique, Paris, 1958).Google Scholar
7. Ono, T., Sur les groupes de Chevalley, J. Math. Soc. Japan 10 (1958), 307–313.Google Scholar
8. Steinberg, R., Automorphisms of finite linear groups, Can. J. Math. 12 (1960), 606–615.Google Scholar
9. Steinberg, R., Lectures on Chevalley groups, mimeographed notes, Yale University, New Haven, Connecticut, 1967-68.Google Scholar