The Klein View of Geometry 
Klein defines a geometry as
Definitions
P and Q have identical gproperties. The study of the geometry defined by G consists of
is consistent. With the axiomatic approach, it is always possible that the axioms lead to a contradiction. In such a case, there can be no "model" for the geometry. As an example, euclidean geometry may be described as having
geometrical property of euclidean geometry. We can use the group to prove that each of Euclid's axioms is a theorem of this geometry. The other advantage of Klein's approach is that it allows us to relate different geometries.
Suppose that G is a group of transformations of the set S, and that H is a subgroup of G.


Theorem K1 Suppose that G is a group of transformations of a set S, and H is a subgroup of G. If two figures are hcongruent, then they are gcongruent.


Proof Suppose that P and Q are hcongruent. Then there is an h in H with h(P) = Q. As H is a subgroup of G, hεG so P and Q are gcongruent.
It follows that each gcongruence class is a union of hcongruenced classes, so that,
Example
Suppose that tεE(2). As t preserves distance, t(C) will be the circle
Now suppose that sεS(2). Since scaling maps circles to circles, we see that
Note that the similarity class of C contains the euclidean class of C.


Theorem K2 Suppose that G is a group of transformations of a set S, and H is a subgroup of G. If D is a gproperty, then it is an hproperty.


Proof Suppose that D is a gproperty, so is invariant under g for all g in G. For h in H, hεG as H≤G, so that D is invariant under h. Thus, the geometry defined by G may have fewer properties.
Example

