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Hurwitz Groups and G2(q)

Published online by Cambridge University Press:  20 November 2018

Gunter Malle
Gunter Malle*
Affiliation:
Mathematisches Institut der Universität Heidelberg, Im Neuenheimer F eld 288, D - 6900 Heidelberg
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Abstract

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Finite factor groups of are called Hurwitz groups. Here we prove that apart from 2G2(3), G2(2), G2(3) and G2(4), all the groups 2G2(32n+1) and G2(q), q = pn, p € P, are Hurwitz groups. For the proof, (2, 3, 7) structure constants are calculated from the character tables [2], [7], and then with the lists of maximal subgroups [8], [5], the existence of generating triples is deduced.

Keywords

20F0530F35
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 33 , Issue 3 , 01 September 1990 , pp. 349 - 357
Copyright
Copyright © Canadian Mathematical Society 1990

References

1. Chang, B., The conjugate classes of Chevalley groups of type (G2), J. Algebra 9 (1968), pp. 190–211.Google Scholar
2. Chang, B., Ree, R., The characters of G2(q), Symposia Mathematica XIII, London 1974, pp. 395–4413.Google Scholar
3. Conder, M., A family of Hurwitz groups with non-trivial centres, Bull. Austr. Math. Soc. 33 (1986), pp. 123–130.Google Scholar
4. Conway, J. H., Curtis, R. T., Norton, S. P., Parker, R. A., Wilson, R. A., Atlas of finite groups, Clarendon Press 1985, Oxford.Google Scholar
5. Cooperstein, B. N., Maximal subgroups of G2(2n), J. Algebra 70 (1981), pp. 23–36.Google Scholar
6. Enomoto, H., The conjugacy classes of Chevalley groups of type (G2) over finite fields of characteristic 2 or 3, J. Fac. Sci. Univ. Tokyo 16 (1970), pp. 497–512.Google Scholar
7. Enomoto, H., The characters of the finite Chevalley groups G2(q), q = 3f, Japan J. Math. N.S. 2 (1976), pp. 191–248.Google Scholar
8. Kleidman, P., The maximal subgroups of the Chevalley groups G2(q) with q odd, the Ree groups G2(q) and of their automorphism groups, J. Algebra 117 (1988), pp. 30–71.Google Scholar
9. Macbeath, A. M., Generators of the linear fractional groups, Proc. Symp. Pure Math. 12 (1969), pp. 14–32.Google Scholar
10. Malle, G., Exceptional groups of Lie type as Galois groups, J. reine angew. Math. 392 (1988), pp. 70–109. Google Scholar
11. Matzat, B. H., Konstruktive Galoistheorie, Springer Lecture Notes 1284, Berlin - Heidelberg - New York 1987.Google Scholar
12. Simpson, W., Frame, J., The character tables for SL3(q\ SU3(q), PSL3(q), U3(q), Can. J. Math. 25 (1973), pp. 486–494.Google Scholar
13. Ward, H. N., On Ree's series of simple groups, Trans. Am. Math. Soc. 121 (1966), pp. 62–89.Google Scholar