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On Free Products of Cyclic Rotation Groups

Published online by Cambridge University Press:  20 November 2018

TH. J. Dekker
TH. J. Dekker*
Affiliation:
University of Amsterdam
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We consider the group of rotations in three-dimensional Euclidean space, leaving the origin fixed. These rotations are represented by real orthogonal third-order matrices with positive determinant. It is known that this rotation group contains free non-abelian subgroups of continuous rank (see 1).

In this paper we shall prove the following conjectures of J. de Groot (1, pp. 261-262):

Theorem 1. Two rotations with equal rotation angles a and with arbitrary but different rotation axes are free generators of a free group, if cos α is transcendental.

Theorem 2. A free product of at most continuously many cyclic groups can be isomorphically represented by a rotation group.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 11 , 1959 , pp. 67 - 69
Copyright
Copyright © Canadian Mathematical Society 1959

References

1. de Groot, J., Orthogonal isomorphic representations of free groups, Can. J. Math., 8 (1956), 256–262.Google Scholar
2. von Neumann, J., Ein System algebraisch unabhangiger Zahlen, Math. Ann., 99 (1928), 134–141.Google Scholar