# geometrical invariants

 The study of a geometry consists largely of identifying and interpreting the geometrical properties, i.e. the invariants of the group defining the geometry. These may be quantities, such as length and angle, or notions such as betweenness and parallelism. In our discussion of the klein view, we noted that, if a group G defines a geometry on the set S, then a subgroup H of G will also define a geometry on S. The two geometries are related. In particular Theorem K2 says that any property of the G-geoemtry will also be a property of the H-geometry. It is sensible, therefore, to establish a property in the largest possible group, and hence obtain the invariance in all the geometries defined by subgroups. We shall illustrate this aspect of the klein view for many of the geometries we have met. An obvious example is the concept of distance in euclidean geometry. We shall use the complex model. Recall that, for t in E(2),    if t is direct, then t(z) = αz + β, with |α| = 1, and    if t is indirect, then t(z) = αz* + β, with |α| = 1. Now consider the quantity |z-w| for complex z and w. If t is direct, then |t(z)-t(w)| = |α(z-w)| = |α||z-w| = |z-w| as |α| = 1. If t is indirect, then |t(z)-t(w)| = |α(z*-w*)| = |α||(z-w)*| = |z-w|. Thus |z-w| is an invariant of euclidean geometry. Of course, it is the usual distance function, but we have proved the invariance directly from the group. The other familiar concepts, such as ratio and angle, can be approached in a similar way, but these are invariants fo larger groups. In keeping with the klein philosophy, we shall prove the invariance in the larger groups.
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