---
tag: 04RG02S6BA
kind: statement
role: proposition
name: Cyclic groups are abelian
msc: [20A05]
uses: [0B85Q41RP6, 00DRK90Z6K, 071DDFVG4J]
sources:
  - url: https://en.wikipedia.org/wiki/Cyclic_group
    locator: Additional properties
---

## Statement

Let $G$ be a [[cyclic group|0KVGH0J5B3]].
Then $G$ is [[abelian|082GHNPJ53]].

## Remarks

If $g$ generates $G$, then any two elements are $g^a$ and $g^b$ for integers $a, b$, and $g^a g^b = g^{a+b} = g^b g^a$ by [[the laws of exponents|00DRK90Z6K]].
The converse is false: the Klein four-group is abelian but not cyclic.
The group $(\mathbb{Q}, +)$ is abelian but not cyclic.

## Explanations

### L1 human

All elements of a cyclic group are powers of one element.
Two powers of the same element always commute, because the exponents just add.
Thus the order of the factors never matters in a cyclic group.