Finite Rotation Groups

Abstract

The special orthogonal group SO ₃ may be identified with the group of rotations of ℝ³ which fix the origin (Chapter 9). If an object is positioned in ℝ³ with its centre of gravity at the origin, then its rotational symmetry group “is” a subgroup of SO ₃ . We are familiar with several possibilities. From a right regular pyramid with an n-sided base we obtain a cyclic group of order n, while a regular plate with n sides exhibits dihedral symmetry and gives D _n . (Regular with two sides means the lens shape described in Exercise 9.12.) In addition, we have the symmetry groups of the regular solids. As we shall see, these are the only possibilities, provided our object has only a finite amount of symmetry. In other words a finite subgroup of SO ₃ is either cyclic, dihedral, or isomorphic to the rotational symmetry group of one of the regular solids. We begin with a less ambitious result which deals with finite subgroups of O ₂.