# Integer Multiples under Addition form Infinite Cyclic Group

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## Theorem

Let $n \Z$ be the set of integer multiples of $n$.

Then $\struct {n \Z, +}$ is a countably infinite cyclic group.

It is generated by $n$ and $-n$:

- $n \Z = \gen n$
- $n \Z = \gen {-n}$

Hence $\struct {n \Z, +}$ can be justifiably referred to as the additive group of integer multiples.

## Proof

From Integer Multiples under Addition form Subgroup of Integers, $\struct {n \Z, +}$ is a subgroup of the additive group of integers $\struct {\Z, +}$.

From Integers under Addition form Infinite Cyclic Group, $\struct {\Z, +}$ is a cyclic group.

So by Subgroup of Cyclic Group is Cyclic, $\struct {n \Z, +}$ is a cyclic group.

The final assertions follow from Subgroup of Infinite Cyclic Group is Infinite Cyclic Group.

$\blacksquare$

## Also see

## Sources

- 1965: J.A. Green:
*Sets and Groups*... (previous) ... (next): $\S 5.4$. Cyclic groups: Example $99$