Editing Translation (geometry) (section)

==As a group==
{{see also|Translation operator (quantum mechanics)#Translation group}}
The set of all translations forms the '''translation group''' <math>\mathbb{T} </math>, which is isomorphic to the space itself, and a [[normal subgroup]] of [[Euclidean group]] <math> E(n) </math>. The [[quotient group]] of <math>E(n) </math> by <math>\mathbb{T} </math> is isomorphic to the group of rigid motions which fix a particular origin point, the [[orthogonal group]] <math> O(n)</math>:
:<math>E(n)/\mathbb{T}\cong O(n) </math>

Because translation is [[commutative]], the translation group is [[Abelian group|abelian]]. There are an infinite number of possible translations, so the translation group is an [[infinite group]].

In the [[theory of relativity]], due to the treatment of space and time as a single [[spacetime]], translations can also refer to changes in the [[Coordinate time|time coordinate]]. For example, the [[Galilean group]] and the [[Poincaré group]] include translations with respect to time.

===Lattice groups===
{{main|Lattice (group)}}

One kind of [[subgroup]] of the three-dimensional translation group are the [[Lattice (group)|lattice groups]], which are [[infinite group]]s, but unlike the translation groups, are [[Finitely generated group|finitely generated]]. That is, a finite [[Generating set of a group|generating set]] generates the entire group.