Euclidean group

Description

The Euclidean group $\mathrm{E}(n)$ is the set of all Euclidean isometries of $\mathbb{R}^n$, with the composition of maps as the operation. It is a group: the identity map is the neutral element, and the inverse of an isometry is an isometry. Each element has the form $x \mapsto A x + b$, with $A$ in the orthogonal group $\mathrm{O}(n)$ and $b \in \mathbb{R}^n$. The direct isometries, with $\det A = 1$, form a subgroup $\mathrm{SE}(n)$ of index $2$. Its elements are the rigid motions.

Remarks

The translations form an abelian normal subgroup $T \cong \mathbb{R}^n$, and $\mathrm{E}(n) / T \cong \mathrm{O}(n)$. The group $\mathrm{E}(n)$ has $n(n+1)/2$ degrees of freedom: $3$ for $n = 2$ and $6$ for $n = 3$. The group is not abelian for $n \geq 2$. For example, a rotation about the origin and a translation do not commute in general. The symmetry group of each figure in $\mathbb{R}^n$ is a subgroup of $\mathrm{E}(n)$.

Explanations

L1 human

The Euclidean group collects all the ways to move space without changing distances. Two such moves done one after the other give again such a move, and each move can be undone. In the plane, three numbers fix a direct move: two for a shift and one for a turn.