Integers Infinite Cyclic Group

Theorem

The Additive Group of Integers $\left({\Z, +}\right)$ 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:

$\forall n \in \Z: n = +^n 1 \in \left \langle {1} \right \rangle$

Thus:

$\left({\Z, +}\right) = \left \langle {1} \right \rangle$

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

$\blacksquare$

Sources

• J.A. Green: Sets and Groups (1965)... (previous)... (next): $\S 5.3$: Example $98$
• Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 25$: Theorem $25.2$
• George McCarty: Topology: An Introduction with Application to Topological Groups (1967): Chapter $\text{II}$
• Allan Clark: Elements of Abstract Algebra (1971)... (previous)... (next): $\S 43$