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 $99$