Translation (geometry)
As a group

The set of all translations forms the translation group T {\displaystyle \mathbb {T} }, which is isomorphic to the space itself, and a normal subgroup of Euclidean group E (n) {\displaystyle E(n)}. The quotient group of E (n) {\displaystyle E(n)} by T {\displaystyle \mathbb {T} } is isomorphic to the group of rigid motions which fix a particular origin point, the orthogonal group O (n) {\displaystyle O(n)}:

Because translation is commutative, the translation group is 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 time coordinate. For example, the Galilean group and the Poincaré group include translations with respect to time.
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