# 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.

$\blacksquare$