# Group of Euclidean motions

From Groupprops

## Definition

The **group of Euclidean motions** in dimensions is defined in the following equivalent ways:

- It is the group of isometries of Euclidean space , where the metric is the Euclidean metric.
- It is the affine orthogonal group of order over the field . In other words, it is the semidirect product of with the orthogonal group , viewed as a subgroup of the general affine group .