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Finite groups generated by reflections, and their subgroups generated by reflections

Published online by Cambridge University Press:  24 October 2008

H. S. M. Coxeter
H. S. M. Coxeter
Affiliation:
Trinity College
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Extract

In connection with his work on singularities of surfaces, Du Val asked me to enumerate certain subgroups in the symmetry groups of the “pure Archimedean” polytopes n21 (n < 5), namely those subgroups which are generated by reflections. For the sake of completeness, I have enumerated such subgroups of all the discrete groups generated by reflections (including the symmetry groups of the regular polytopes). The work involved being somewhat intricate, several slips would have been overlooked but for the information that Du Val was able to supply from the (apparently remote) theory of surfaces.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 30 , Issue 4 , October 1934 , pp. 466 - 482
Copyright
Copyright © Cambridge Philosophical Society 1934

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References

* 2, 595.

† 2, 601.

‡ 2, 619.

* More generally, [3p+q]=[3o,p,q].

† 2, 610.

* 3, § 1.

† 3, § 5.

* 1, 336.

† E.g., a pentagram ().

* 1, 346 (bottom of page).

* 1, 365.

* 1, 366. This polytope is called t 0,na n+1 in 3; Mrs Stott calls it e nS (n + 2). The vertices (1, −1; 0n) of the are transposed by reflection in the prime x 1 = x 2; such reflections generate [3n] (see 2, 606).

† r −1 = ½ (√ 5 − 1).

* Val, Du, Proc. Camb. Phil. Soc. 30 (1934), 460–465.CrossRefGoogle Scholar

* The symbol ½ (±) indicates that we must take either every combination of an even number of negative signs or every combination of an odd number.

† Cf. Littlewood, D. E., “The groups of the regular solids in n dimensions”, Proc. London Math. Soc. (2), 32 (1931), 13Google Scholar; Robinson, G. de B., “On the orthogonal groups in four dimensions”, Proc. Camb. Phil. Soc. 27 (1931), 43.CrossRefGoogle Scholar

‡ r = ½ (√ 5 + 1) The dash indicates even permutation.

* We may take ea 7 in the prime ∑x = 0, using either the “even” or the “odd” coordinates for 421.

* When n = 2, 2 {4} √2.