On Free Products of Cyclic Rotation Groups

Abstract

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.