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Abstract definitions for the symmetry groups of the regular polytopes, in terms of two generators. Part II: the rotation groups

Published online by Cambridge University Press:  24 October 2008

H. S. M. Coxeter
H. S. M. Coxeter
Affiliation:
Trinity College
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The complete (or “extended”) symmetry groups, investigated in Part I, are certain groups of orthogonal transformations, generated by reflections. Every such group has a subgroup of index two, consisting of those transformations which are of positive determinant (i.e., “movements” or “displacements”). The positive subgroup (in this sense) of [k1, k2, …, kn−1] is denoted by [k1, k2, …, kn−1]′, and is “the rotation group” (or, briefly, “the group”) of either of the regular polytopes {k1, k2, …, kn−1}, {kn−1, kn−2, …, k1}; e.g., [3, 4]′ is the octahedral group.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 33 , Issue 3 , July 1937 , pp. 315 - 324
Copyright
Copyright © Cambridge Philosophical Society 1937

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References

* Coxeter, and Todd, , Proc. Camb. Phil. Soc. 32 (1936), 194–200.CrossRefGoogle Scholar

† Part I, 194.

‡ Klein, F., Vorlesungen über das Ikosaeder (Leipzig, 1884), 16, 19.Google Scholar

§ Todd, J. A., Proc. Camb. Phil. Soc. 27 (1931), 218.Google Scholar Cf. Littlewood, D. E., Proc. London Math. Soc. (2), 32 (1930), 18, 14,Google Scholar where six generators are used.

∥ Todd, loc. cit. 228. Hereafter (as in Part I), we shall refer to that paper as G.S.R.P.

* Part I, 197.

† Artin, E., “Theorie der Zöpfe”, Abhandl. Math. Sem. Hamburg. Univ. 4 (1926), 52–54.Google Scholar Artin's definition is actually

for the symmetric group of degree n. ([3n−1] is the symmetric group of degree n + 1)

* Coxeter, , Journal London Math. Soc. 11 (1936), 151.Google Scholar There we showed that the relations

imply

whence

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† G.S.R.P. 226.

‡ Part I, 199. A printer's error should be corrected in the re-statement of this definition on the eighth line of p. 195.

§ Cf. Coxeter, , Proc. London Math. Soc. (2), 41 (1936), 289Google Scholar (6·8). We have changed S into S −1, for the sake of analogy with (12). The relation R 2m = 1 is clearly superfluous. When n is odd, these relations define the hyper-pyritohedral group [(3n−2)′, 4].

* Coxeter, loc. cit. 287 (5·9).

† G.S.R.P. 216, 225.

‡ G.S.R.P. 224, 217.

* Coxeter and Sinkov, “The groups determined by the relations

S l = T m = (S −1T −1ST)p = 1”,

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† G.S.R.P. 229. The first sentence should read, “The group contains a sub-group of order two, generated by the central inversion; since this operation…”.

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‡ “The groups of the regular solids in n dimensions”, Proc. London Math. Soc. (2), 32 (1930), 12.Google Scholar This is essentially the same as the decomposition of an orthogonal substitution by Goursat, E., Ann. Sci. de l'Éc. Norm. Sup. (3), 6 (1889), 28.Google Scholar

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§ G.S.R.P. 226.

∥ Coxeter, , Annals of Math. 35 (1934), 608CrossRefGoogle Scholar (vii). In G.S.R.P. 230, Todd expresses S as a permutation.

* The third analogous result is that two binary tetrahedral groups with a common central lead to the group [3, 4, 3]″, of order

† G. de B. Robinson, loc. cit.

‡ Coxeter, , Proc. London Math. Soc. (2), 38 (1935), 331.Google Scholar

§ Math. Ann. 104 (1930), 64.Google Scholar