Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-20T02:40:55.639Z Has data issue: false hasContentIssue false
  • English
  • Français

Some Two-Dimensional Unitary Groups Generated by Three Reflections

Published online by Cambridge University Press:  20 November 2018

D. W. Crowe
D. W. Crowe*
Affiliation:
University College, Ibadan
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.

Shephard and Todd (5) give generators for the finite primitive irreducible groups generated by two unitary reflections in U2. It is the purpose of the present paper to give generating reflections, and defining relations in terms of these reflections, for the seven such groups requiring three generating reflections, that is, for their nos. 7, 11, 12, 13, 15, 19, 22. The reflections are chosen whenever possible so that their product has the property suggested by Theorem 5.4 of (5). That is, except for no. 15, the period of the product of the three generating reflections is h = m2 + 1, and the characteristic roots of this product are 2πim1/h and 2πim2/h, where m1 and m2 are the “exponents“ (5, p. 282) of the group. The reason for the impossibility of such a choice for no. 15 is given in § 4. In § 5 the homomorphisms between these groups and certain groups of motions in elliptic 3-space are determined.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 13 , 1961 , pp. 418 - 426
Copyright
Copyright © Canadian Mathematical Society 1961

References

1. Coxeter, H. S. M., Groups generated by unitary reflections of period two, Can. J. Math., 9 (1957), 243–272.Google Scholar
2. Coxeter, H. S. M. and Moser, W. O. J., Generators and reflections for discrete groups (Berlin, 1957).Google Scholar
3. Crowe, D. W., The groups of regular complex polygons, Can. J. Math., 13 (1961), 149–156.Google Scholar
4. Goursat, E., Sur les substitutions orthogonales et les divisions régulières de l'espace, Ann. Sci. Ecole Norm. Sup. (3), 6 (1889), 9–102.Google Scholar
5. Shephard, G. C. and Todd, J. A., Finite unitary reflection groups, Can. J. Math., 6 (1954), 274–304.Google Scholar