Additive Groups of Integers and Integer Multiples are Isomorphic

Theorem

Let $n \in \Z_{> 0}$ be a strictly positive integer.

Let $\struct {n \Z, +}$ denote the additive group of integer multiples.

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

Then $\struct {n \Z, +}$ is isomorphic to $\struct {\Z, +}$.

Proof

We have that:

Infinite Cyclic Group is Isomorphic to Integers.

Integer Multiples under Addition form Infinite Cyclic Group.

Infinite Cyclic Group is Unique up to Isomorphism

Hence the result.

$\blacksquare$

Sources

• 1965: J.A. Green: Sets and Groups ... (previous) ... (next): Chapter $7$: Homomorphisms: Exercise $5$