Endomorphism from Integers to Multiples

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

Let $\phi: \struct {\Z, +} \to \struct {\Z, +}$ be a mapping.

Then $\phi$ is a group endomorphism :
 * $\exists k \in \Z: \forall n \in \Z: \map \phi n = k n$

Necessary Condition
Let $\phi: \struct {\Z, +} \to \struct {\Z, +}$ be an endomorphism.

Let $k = \map \phi 1$.

We have that $n = \overbrace {1 + \cdots + 1}^n$ for any positive integer $n$.

Thus:

Also:

Thus:
 * $\forall n \in \Z: \map \phi n = k n$

Sufficient Condition
Let $k \in \Z$ such that:
 * $\forall n \in \Z: \map \phi n = k n$

Then:

Thus $\phi: \struct {\Z, +} \to \struct {\Z, +}$ is a group homomorphism from $\Z$ to $\Z$.

Hence by definition $\phi$ is a group endomorphism.