# Definition:Translation Mapping/Abelian Group

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## Definition

Let $\struct {G, +}$ be an abelian group.

Let $g \in G$.

Then **translation by $g$** is the mapping $\tau_g: G \to G$ defined by:

- $\forall h \in G: \map {\tau_g} h = h + \paren {-g}$

where $-g$ is the inverse of $g$ with respect to $+$ in $G$.

## Caution

It is easy to confuse the mappings $\tau_{\mathbf x}$ and $\tau_{-\mathbf x}$, and the choice made here is arbitrary.

The map $\tau_{\mathbf x}$ can be understood (appealing to our planar $\R^2$ intuition) as translating the coordinate axes by $\mathbf x$.

## Also see

- Results about
**translation mappings**can be found here.