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In Klein's view a geometry consists of a set *S* and a subgroup *G* of the group *Bij*(*S*) of all bijections from *S* to itself.

The prototype example is the case *S* = **R**^{2} and then, for example, the group associated with plane Euclidean geometry is the *group of rigid motions of the plane*. (See example 1. below.)

The elements of *S* are called *points* and we think of the elements of *G* as acting on *S* by mapping points to points.

Two subsets of *S* are regarded as *equivalent* or "the same" if there is an element of *G* taking one set into the other.

In general, we endow our set *S* with *extra structure* and look at the subgroup of *Bij*(*S*) which *preserves this structure*.

**Examples**

- In the Euclidean plane the extra structure is the usual metric
*d*which measures the distance between points in the plane.

We define*d*((*x*_{1},*y*_{1}), (*x*_{2},*y*_{2})) = √[(*x*_{1}-*x*_{2})^{2}+ (*y*_{1}-*y*_{2})^{2}].

The group which preserves this is the set of transformations which map pairs of points to points the same distance apart. Such elements*g*satisfy

*d*(*g*(**x**),*g*(**y**)) =*d*(**x**,**y**)

- and such transformations are called
*isometries*or*rigid motions*.

For plane Euclidean geometry equivalence is called*congruence*. For example, two triangles are congruent if their corresponding sides have equal length (or equivalently, two sides and the "included angle", ... ).- We may take a metric on
*S*and then consider the group of bijections from*S*to itself which preserve this structure by being*continuous*(and having continuous inverse maps). The geometry associated with this group is then called a*topology*. For those who have met this idea before, the transformations are called*homeomorphisms*or*topological isomorphisms*.- If we take the set
*S*to be the plane**R**^{2}and we take the group of bijections on*S*which*preserve angles*, we get a group whose associated geometry is called*similarity geometry*. The concept of equivalence is then called*similarity*. For example, two triangles which are*similar*have their angles the same (and their sides are then "in proportion").

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