Cyclic groups are abelian

Statement

Let $G$ be a cyclic group. Then $G$ is abelian.

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. 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.