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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