E(n) - Knowlpedia

For every integer $n\ge1$ , $\mathrm{E}(n)$ is the group of Euclidean isometries of $\mathbb R^{n}$ . Each element is $x\mapsto Ax+b$ with $A\in\mathrm{O}(n,\mathbb R)$ and $b\in\mathbb R^{n}$ , and

(A,b)(A^{\prime},b^{\prime})=(AA^{\prime},Ab^{\prime}+b).

Dimensions and structure

This group has real dimension $n(n+1)/2$ . This gives the semidirect product of translations by the indicated orthogonal group.

Direct verification

The displayed product is composition of affine maps. Orthogonal matrices preserve the Euclidean distance, and any Euclidean isometry is a translation followed by an orthogonal map. Adding the translation dimension to n(n−1)/2 gives n(n+1)/2.