Integers under Addition form Infinite Cyclic Group

Theorem
The additive group of integers $\struct {\Z, +}$ is an infinite cyclic group which is generated by the element $1 \in \Z$.

Proof
By Epimorphism from Integers to Cyclic Group and integer multiplication:


 * $\forall n \in \Z: n = \map {+^n} 1 \in \gen 1$

Thus:
 * $\struct {\Z, +} = \gen 1$

and thus, by the definition of a cyclic group, is cyclic.