---
tag: 02B1T50JD1
kind: example
name: Infinite cyclic group
msc: [20A05]
instance_of: [0KVGH0J5B3]
uses: [0B85Q41RP6, 0BSCJA5G53]
sources:
  - url: https://en.wikipedia.org/wiki/Cyclic_group
    locator: Integer and modular addition
  - url: https://en.wikipedia.org/wiki/Cyclic_group
    locator: Definition and notation
---

## Example

The [[additive group of integers|0BSCJA5G53]] $(\mathbb{Z}, +)$ is an infinite [[cyclic group|0KVGH0J5B3]].
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.