Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-25T05:17:53.329Z Has data issue: false hasContentIssue false
  • English
  • Français

Generators for Simple Groups

Published online by Cambridge University Press:  20 November 2018

Robert Steinberg
Robert Steinberg*
Affiliation:
University of California
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.

The list of known finite simple groups other than the cyclic, alternating, and Mathieu groups consists of the classical groups which are (projective) unimodular, orthogonal, symplectic, and unitary groups, the exceptional groups which are the direct analogues of the exceptional Lie groups, and certain twisted types which are constructed with the aid of Lie theory (see §§3 and 4 below). In this article, it is proved that each of these groups is generated by two of its elements. It is possible that one of the generators can be chosen of order 2, as is the case for the projective unimodular group (1), or even that one of the generators can be chosen as an arbitrary element other than the identity, as is the case for the alternating groups. Either of these results, if true, would quite likely require methods much more detailed than those used here.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 14 , 1962 , pp. 277 - 283
Copyright
Copyright © Canadian Mathematical Society 1962

References

1. Albert, A. A. and Thompson, J., Illinois J. Math., 3 (1959), 421.Google Scholar
2. Chevalley, C., Sur certains groupes simples, Tôhoku Math. J., 7 (1955), 14.Google Scholar
3. Hertzig, D., On simple algebraic groups, Short communications, Int. Cong. Math. (Edinburgh, 1958).Google Scholar
4. Ree, R., A family of simple groups associated with the simple Lie algebra of type G%, Bull. Amer. Math. Soc, 66 (1960), 508.Google Scholar
5. Chevalley, Séminaire C., Classification des groupes de Lie algébriques (Paris, 1956-8).Google Scholar
6. Steinberg, R., Finite reflection groups, Trans. Amer. Math. Soc, 91 (1959), 493.Google Scholar
7. Steinberg, R., Variations on a theme of Chevalley, Pacific J. Math., 9 (1959), 875.Google Scholar
8. Steinberg, R., The simplicity of certain groups, Pacific J. Math., 10 (1960), 1039.Google Scholar
9. Steinberg, R., Automorphisms of finite linear groups, Can. J. Math., 12 (1960), 606.Google Scholar
10. Suzuki, M., A new type of simple groups of finite order, Proc. Nat. Acad. Sci., 46 (1960), 868.Google Scholar
11. Tits, J., Les “formes réelles'” des groupes de type E, Séminaire Bourbaki, Exposé 162 (Paris, 1958).Google Scholar
12. Tits, J., Sur la trialité et certains groupes qui s'en déduisent, Publ. Math. Inst. Hautes Etudes Sci., 2 (1959), 14.Google Scholar
13. Room, T. G. and Smith, R. J., A generation of the symplectic group, Quart. J. Math., 9 (1958), 177.Google Scholar
14. Room, T. G., The generation by two operators of the symplectic group over GF﹛2), J. Austr. Math. Soc, 1 (1959), 38.Google Scholar
15. Stanek, P. F. G., Two element generation of the symplectic group, Bull. Amer. Math. Soc, 67 (1961), 225.Google Scholar