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.

Proof

Proof of Injectivity

\(\ds \forall h_1, h_2 \in G: \, \)

\(\ds \map {\tau_g} {h_1}\)

\(=\)

\(\ds \map {\tau_g} {h_2}\)

\(\ds \leadsto \ \ \)

\(\ds h_1 + \paren {-g}\)

\(=\)

\(\ds h_2 + \paren {-g}\)

Definition of $\tau_g$

\(\ds \leadsto \ \ \)

\(\ds h_1\)

\(=\)

\(\ds h_2\)

Cancellation Laws

$\Box$

Proof of Surjectivity

\(\ds \forall h_1 \in G: \exists h_2 \in G: \, \)

\(\ds h_1 + g\)

\(=\)

\(\ds h_2\)

\(\ds \leadsto \ \ \)

\(\ds h_1\)

\(=\)

\(\ds h_2 + \paren {-g}\)

\(\ds \leadsto \ \ \)

\(\ds h_1\)

\(=\)

\(\ds \map {h_2} g\)

Definition of $\tau_g$

$\blacksquare$