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Automorphisms of Finite Linear Groups

Published online by Cambridge University Press:  20 November 2018

Robert Steinberg
Robert Steinberg*
Affiliation:
University of California, Los Angeles
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By the methods used heretofore for the determination of the automorphisms of certain families of linear groups, for example, the (projective) unimodular, orthogonal, symplectic, and unitary groups (7, 8), it has been necessary to consider the various families separately and to give many case-by-case discussions, especially when the underlying vector space has few elements, even though the final results are very much the same for all of the groups. The purpose of this article is to give a completely uniform treatment of this problem for all the known finite simple linear groups (listed in §2 below). Besides the * ‘classical groups” mentioned above, these include the “exceptional groups,” considered over the complex field by Cartan and over an arbitrary field by Dickson, Chevalley, Hertzig, and the author (3, 4, 5, 6, 10, 15). The automorphisms of the latter groups are given here for the first time.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 12 , 1960 , pp. 606 - 615
Copyright
Copyright © Canadian Mathematical Society 1960

References

1. Artin, E., Orders of classical simple groups, Comm. Pure Appl. Math., 8 (1955), 455.Google Scholar
2. Bruhat, F., Représentations induites des groupes de Lie semi-simples connexes, C. R. Acad. Sci. Paris, 235 (1954), 437.Google Scholar
3. Cartan, E., Oeuvres complètes (Paris, 1952).Google Scholar
4. Chevalley, C., Sur certains groupes simples, Tôhoku Math., 7 (1955), 14.Google Scholar
5. Dickson, L.E., Linear groups (Leipzig, 1901).Google Scholar
6. Dickson, L.E., A new system of simple groups, Math. Ann., 60 (1905), 137.Google Scholar
7. Dieudonné, J., On the automorphisms of the classical groups, Mem. Amer. Math. Soc, 2 (1951).Google Scholar
8. Dickson, L.E., La géométrie des groupes classiques, Ergeb. der Math. u. i. Grenz. (1955).Google Scholar
9. Dynkin, E.B., The structure of semi-simple algebras, Amer. Math. Soc. Translation No. 17.Google Scholar
10. Hertzig, D., On simple algebraic groups, Short communications, Int. Congress of Math. (Edinburgh, 1958).Google Scholar
11. Ree, R., On some simple groups defined by C. Chevalley, Trans. Amer. Math. Soc, 84 (1957), 392.Google Scholar
12. Schreier, O. and van der Waerden, B. L., Die Automorphismen der projektiven Gruppen, Abh. Math. Sem. Univ. Hamburg, 6 (1928), 303.Google Scholar
13. Chevalley, C. Séminaire, Classification des Groupes de Lie Algébriques (Paris, 1956-8).Google Scholar
14. Séminaire, “Sophus Lie” (Paris, 1954-1955).Google Scholar
15. Steinberg, R., Variations on a theme of Chevalley, Pac. J. Math., 9 (1959), 875.Google Scholar
16. Weyl, H., The structure and representations of continuous groups, I. A. S. notes (Princeton, 1934-1935).Google Scholar