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Small rank exceptional Hurwitz groups. (English) Zbl 0861.20017
Kantor, William M. (ed.) et al., Groups of Lie type and their geometries. Proceedings of the conference held in Como, Italy, June 14-19, 1993. Cambridge: Cambridge University Press. Lond. Math. Soc. Lect. Note Ser. 207, 173-183 (1995).
The simple exceptional groups of Lie type \(^3D_4(q)\) and \(^2F_4(2^{2n+1})'\) are shown to be generated by an element of order 2 and one of order 3. Moreover, these generators can be chosen to have a product of order 7 (Hurwitz group) iff \(p\neq 3\), \(q\neq 4\) in the case \(^3D_4(q)\), \(q=p^n\) and \(n\equiv 1\pmod 3\) in the case \(^2F_4(2^{2n+1})'\). As with similar studies with other groups of Lie type the proof depends on use of detailed information on maximal subgroups and generic character tables in a form which allows the calculation of normalized structure constants of triples of conjugacy classes (CHEVIE system). The study of \((2,3)\)-generation of exceptional groups of Lie type is completed in joint work with F. Lübeck (to appear J. Lond. Math. Soc.).
The following references have appeared: [3] Commun. Algebra 22, No. 4, 1321-1347 (1994; Zbl 0805.20038) and 24, No. 2, 487-515 (1996; Zbl 0847.20043); [5] Appl. Algebra Eng. Commun. Comput. 7, No. 3, 175-210 (1996; Zbl 0847.20006); [13] J. Algebra 168, No. 1, 353-370 (1994; Zbl 0819.20053).
For the entire collection see [Zbl 0830.00029].

MSC:
20D06 Simple groups: alternating groups and groups of Lie type
20C40 Computational methods (representations of groups) (MSC2010)
20F05 Generators, relations, and presentations of groups
Software:
CHEVIE
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