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. ## Subgroup structure
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 *u*^{−1}*tu* 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.
## Related topics |