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In mathematics, the Euclidean group is the symmetry group associated with Euclidean geometry. It is therefore one of the oldest and most studied groups, at least in the cases of dimension 2 and 3 — implicitly, many of its properties are familiar, if not in mathematical language. Writing E(n) for the Euclidean group of symmetries of n-dimensional Euclidean space, it may also be described as the isometry group of the Euclidean metric. It has dimension
which gives 3 in case n = 2, and 6 for n = 3.
The Euclidean group has as subgroups the group T of translations, and the orthogonal group O(n). Any element of E(n) is a product of a translation followed by an orthogonal transformation, in a unique way. From the point of view of group theory, one notices that T is a normal subgroup of E(n): for any translation t and any isometry u, we have
again a translation (one can say, through a displacement that is u acting on the displacement of t).
Now SO(n), the special orthogonal group, is a subgroup of O(n), of index two. Therefore E(n) has a subgroup E+(n), also of index two, consisting of direct isometries. That is, isometries not involving a change of orientation; equally, those represented as a translation followed by a rotation, rather than a translation followed by some kind of reflection (in dimensions 2 and 3, these are the familiar reflections in a mirror line or plane, which may be taken to include the origin).
Relation to the affine group
The Euclidean group E(n) is a subgroup of the affine group for n dimensions, and in such a way as to respect the semidirect product structure of both groups. As a consequence, Euclidean group elements can also be represented as square matrices of size n + 1, as explained for the affine group.
In the terms of the Erlangen programme, Euclidean geometry is therefore a specialisation of affine geometry. All affine theorems apply; the extra factor is the notion of distance, from which angle can be deduced.
Rigid body motions
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