Group homomorphism/Z to group/Fact/Proof

Proof

Let ${}g\in G$ be fixed. That the mapping

$\varphi _{g}\colon \mathbb {Z} \longrightarrow G,n\longmapsto g^{n},$

is a group homomorphism, is just a reformulation of the exponential laws. Because of ${}\varphi _{g}(1)=g^{1}=g$, we obtain from the power mapping ${}\varphi _{g}$ the group element back. Moreover, a group homomorphism ${}\varphi \colon \mathbb {Z} \rightarrow G$ is uniquely determined by ${}\varphi (1)$, as ${}\varphi (n)=(\varphi (1))^{n}$ for ${}n$ positive, and ${}\varphi (n)={\left((\varphi (1))^{-1}\right)}^{-n}$ for ${}n$ negative must hold.

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