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