Group of Euclidean motions

Definition
The group of Euclidean motions in $$n$$ dimensions is defined in the following equivalent ways:


 * It is the group of isometries of Euclidean space $$\R^n$$, where the metric is the Euclidean metric.
 * It is the affine orthogonal group of order $$n$$ over the field $$\R$$. In other words, it is the semidirect product of $$\R^n$$ with the orthogonal group $$O(n,\R)$$, viewed as a subgroup of the general affine group $$GA(n,\R)$$.