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A Presentation of the Groups PSL(2, p) with Three Defining Relations

Published online by Cambridge University Press:  20 November 2018

Hans J. Zassenhaus
Hans J. Zassenhaus*
Affiliation:
The Ohio State University, Columbus, Ohio
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H. Behr and J. Mennicke (1) have proven that the group PSL(2, p) can be presented by the following system of generators and relations:

1

From this presentation, it follows that the three relations

2

for the same generators S and T suffice if p > 3, p ≠ 17. If p = 3, it is well known that the relations S3 = 1, T2 = 1, and (ST)3 = 1 define PSL(2, 3). For p = 2, the relations S3 = 1, T2 = 1, and (ST)2 = 1 define PSL(2, 2). For p = 17, the three relations

3

will suffice.

Indeed, the group G, with generators S, T and defining relations (2), contains the subgroup 〈Sp〉 in its centre.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 21 , 1969 , pp. 310 - 311
Copyright
Copyright © Canadian Mathematical Society 1969

References

1. Behr, H. and Mennicke, J., A presentation of the groups PSL(2, p), Can. J. Math. 20 (1968), 1432–1438.Google Scholar
2. Schur, I., Untersuchungen iiber die Darstellungen der endlichen Gruppen durch gebrochene lineare Substitutionen, J. Reine Angew. Math. 182 (1907), 85–137.Google Scholar