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A **Geometry** consists of a set *S* and a subgroup *G* of the group *Bij*(*S*) of all bijections from *S* to itself.

For Euclidean geometry, *S* = **R**^{n} and the group is the set *I*(**R**^{n}) of all length preserving maps or **isometries**.

Every element of *I*(**R**^{n}) is of the form *T*_{a} ∘ *L* with *T*_{a} a translation ** x** ↦

Then

If f is a symmetry of the line: *f* ∈ *I*(**R**), then *f*(*x*) = *a* + *x* or *f*(*x*) = *a* - *x* and *f* is either a translation or reflection (in a point).

*Direct* isometries of the plane: *I*(**R**^{2}) are either *rotations* (about a point) or *translations*.

*Opposite* symmetries of **R**^{2} are either *reflections* (in a line) or *glides* (a reflection in a line followed by translation parallel to the line).

*Direct* isometries of **R**^{3} are either *rotations* (about an *axis* in **R**^{3}) or *translations* or *screws* (a rotation about a line followed by translation parallel to the line).

*Opposite* symmetries of **R**^{3} are either *reflections* (in a plane) or *glides* (a reflection in a plane followed by translation parallel to the plane) or *rotatory reflections* (rotaion about a line followed by reflection in a plane perpendicular to the line).

A symmetry group of a figure F ⊆ **R**^{n} is the set of all symmetries *f* in *I*(**R**^{n}) for which *f*(*F*) = *F*.

Finite subgroups of *I*(**R**^{2}) are isomorphic either to *C*_{n}: a cyclic group of order *n* (generated by a rotation by 2*π* /_{n} about some point) or *D*_{n}: a dihedral group of order 2*n* (generated either by a similar rotation and one reflection in a line through the point, or by two such reflections) and this classifies the subgroups up to conjugacy inside the group *I*(**R**^{2}).

Finite subgroups of *I*(**R**^{3}) which consist only of direct symmetries are isomorphic either to *C*_{n} (generated by a rotation by 2*π* /_{n} about some axis) or *D*_{n} (a dihedral group of order 2*n* consisting of rotations as in the *C*_{n} case and rotations by π) or to the rotational symmetry groups of one of the Platonic solids (*A*_{4} , *S*_{4} or *A*_{5}).

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