Mapping from Additive Group of Integers to Powers of Group Element is Homomorphism

Theorem

Let $\struct {G, \circ}$ be a group.

Let $g \in G$.

Let $\struct {\Z, +}$ denote the additive group of integers.

Let $\phi_g: \struct {\Z, +} \to \struct {G, \circ}$ be the mapping defined as:

$\forall k \in \Z: \map {\phi_g} k = g^k$

Then $\phi_g$ is a (group) homomorphism.

Proof

Let $k, l \in \Z$.

\(\ds \map {\phi_g} {k + l}\)

\(=\)

\(\ds a^{k + l}\)

\(\ds \)

\(=\)

\(\ds a^k a^l\)

\(\ds \)

\(=\)

\(\ds \map {\phi_g} k \circ \map {\phi_g} l\)

thus proving that $\phi_g$ is a homomorphism as required.

$\blacksquare$

Sources

• 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 7.1$. Homomorphisms: Example $129$