Translation Mapping is Bijection

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

Let $g \in G$.

Let $\tau_g: G \to G$ be the translation by $g$:


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

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

Then $\tau_g$ is a bijection.