Summary of Results

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 = Rⁿ and the group is the set I(Rⁿ) of all length preserving maps or isometries.

Every element of I(Rⁿ) is of the form T_a ∘ L with T_a a translation x ↦ a + x and L a length preserving linear map ∈ O(n).
Then det(L) = ±1. If det(L) = +1 the isometry is direct, otherwise it is opposite.

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(R²) are either rotations (about a point) or translations.
Opposite symmetries of R² are either reflections (in a line) or glides (a reflection in a line followed by translation parallel to the line).

Direct isometries of R³ are either rotations (about an axis in R³) or translations or screws (a rotation about a line followed by translation parallel to the line).
Opposite symmetries of R³ 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 ⊆ Rⁿ is the set of all symmetries f in I(Rⁿ) for which f(F) = F.

Finite subgroups of I(R²) are isomorphic either to C_n: a cyclic group of order n (generated by a rotation by 2π /_n about some point) or D_n: a dihedral group of order 2n (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(R²).

Finite subgroups of I(R³) which consist only of direct symmetries are isomorphic either to C_n (generated by a rotation by 2π /_n about some axis) or D_n (a dihedral group of order 2n 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 (A₄ , S₄ or A₅).