Finitely generated group

Definition

A group $G$ is finitely generated when there is a finite subset $S$ of $G$ with $\langle S \rangle = G$, where $\langle S \rangle$ is the subgroup generated by $S$.

Remarks

Each finite group is finitely generated, because it is generated by the finite set of all its elements. The group $(\mathbb{Z}^n, +)$ is finitely generated by the $n$ standard basis vectors. The group $(\mathbb{Q}, +)$ is not finitely generated: the subgroup generated by finitely many fractions has a common denominator, and it does not contain all fractions.

Explanations

L1 human

A group is finitely generated when a finite list of elements is enough to build all elements. The group may still be infinite, as the integers show: one generator $1$ gives all integers. The rational numbers need infinitely many generators.