The substance of this chapter can be expressed as the slogan

Group theory is geometry and geometry is group theory.

In other words, every group is a transformation group: the only purpose of being a group is to act on a space. Conversely, geometry can be discussed in terms of transformation groups. Given a space X and a group G made up of transformations of X, the geometric notions are quantities measured on X which are invariant under the action of G. This chapter formalises the relation between geometry and groups, and discusses some geometric issues for which group theory is a particularly appropriate language.

The action of a transformation group on a space is another way of saying symmetry. To say that an object has symmetry means that it is taken into itself by a group action: rotational symmetry means symmetry under the group of rotations about an axis. As a frivolous example, Coventry market pictured in Figure 6.0 has (approximate) rotational symmetry: if you stand at the centre, all directions outwards are virtually indistinguishable; you can understand a coordinate frame as a signpost to break the symmetry, and to enable people to find their way around.

Each of the geometries studied in previous chapters had transformations associated with it: Euclidean motions of E2, orthogonal transformations as motions of S2, Lorentz transformations as motions of H2, and affine and projective linear transformations of An and ℙn.