Integer Multiples under Addition form Infinite Cyclic Group

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

• Definition:Additive Group of Integer Multiples

Sources

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