Integer Multiples under Addition form Infinite Cyclic Group

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

Then $\left({n \Z, +}\right)$ is a countably infinite cyclic group.

It is generated by $n$ and $-n$:
 * $n \Z = \left \langle n \right \rangle$
 * $n \Z = \left \langle {-n} \right \rangle$

Group Axioms
Taking the group axioms in turn:

G0: Closure
$\left({n \Z, +}\right)$ is closed under $+$ by Integer Multiples Closed under Addition.

G1: Associativity
$+$ is associative in $\left({n \Z, +}\right)$ by dint of associativity of integer addition.

G2: Identity
$0 = n 0 \in n \Z$, so $\left({n \Z, +}\right)$ has an identity.

G3: Inverses
Let $x \in n \Z$.

Then $\exists p \in \Z: x = n p$ and so $-x = n \left({-p}\right) \in \left({n \Z, +}\right)$.

Commutativity
$+$ is commutative in $\left({n \Z, +}\right)$ by dint of commutativity of integer addition.

Countably Infinite
From the description of the set of integer multiples:
 * $n \Z = \left\{{\ldots, -3n, -2n, -n, 0, n, 2n, 3n, \ldots}\right\}$

there is an obvious bijection to the set of integers $\Z$, which is itself a countably infinite set.

Cyclic
Its cyclic nature is shown in Subgroup of Infinite Cyclic Group.