Endomorphism from Integers to Multiples
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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 if and only if:
- $\exists k \in \Z: \forall n \in \Z: \map \phi n = k n$
Proof
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:
\(\ds \map \phi n\) | \(=\) | \(\ds \map \phi {\overbrace {1 + \cdots + 1}^n}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \overbrace {\map \phi 1 + \cdots + \map \phi 1}^n\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \overbrace {k + \cdots + k}^n\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds k n\) |
Also:
\(\ds \map \phi 1\) | \(=\) | \(\ds \map \phi {1 + 0}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \map \phi 1 + \map \phi 0\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map \phi 0\) | \(=\) | \(\ds 0\) | |||||||||||
\(\ds \) | \(=\) | \(\ds k \cdot 0\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map \phi {-1}\) | \(=\) | \(\ds -k\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \forall n \in \N_{>0}: \, \) | \(\ds \map \phi {-n}\) | \(=\) | \(\ds -k n\) | similar with above |
Thus:
- $\forall n \in \Z: \map \phi n = k n$
$\Box$
Sufficient Condition
Let $k \in \Z$ such that:
- $\forall n \in \Z: \map \phi n = k n$
Then:
\(\ds \forall n, m \in \Z: \, \) | \(\ds \map \phi {n + m}\) | \(=\) | \(\ds k \paren {n + m}\) | |||||||||||
\(\ds \) | \(=\) | \(\ds k n + k m\) | Integer Multiplication Distributes over Addition | |||||||||||
\(\ds \) | \(=\) | \(\ds \map \phi n + \map \phi m\) |
Thus $\phi: \struct {\Z, +} \to \struct {\Z, +}$ is a group homomorphism from $\Z$ to $\Z$.
Hence by definition $\phi$ is a group endomorphism.
$\blacksquare$
Sources
- 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{II}$: Groups: Exercise $\text{L}$
- 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: Group Homomorphism and Isomorphism: $\S 61$