Infinite cyclic group

Example

The additive group of integers $(\mathbb{Z}, +)$ is an infinite cyclic group. The element $1$ generates it, because each integer is a sum of copies of $1$ or of $-1$. Its only generators are $1$ and $-1$.

Remarks

If $G = \langle g \rangle$ is an infinite cyclic group, then the powers $g^k$ for $k \in \mathbb{Z}$ are all distinct. Then $k \mapsto g^k$ is an isomorphism from $\mathbb{Z}$ to $G$, thus each infinite cyclic group is isomorphic to $(\mathbb{Z}, +)$. The group of the translations of the plane by integer multiples of a fixed vector is an infinite cyclic group.

Explanations

L1 human

In an infinite cyclic group, the powers of the generator never repeat. You can walk forever in both directions without coming back. This is exactly how the integers behave under addition.