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
- n(n + 1)/2
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).
Together, these facts imply that E(n) is the semidirect product of O(n) extended by T. In other words O(n) is (in the natural way) also the quotient group of E(n) by 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
Another use of a Euclidean group is for the kinematics of a rigid body, in classical mechanics. A rigid body motion is in effect the same as a curve in E+(3).
The Euclidean groups are Lie groups, so that calculus notions can be adapted immediately from this setting.