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  "body": "\n## Definition\n\nA [[group|G5S1M9]] $G$ 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 $S$|0RP7VWQ0YN]].\n\n## Remarks\n\nEach finite group is finitely generated, because it is generated by the finite set of all its elements.\nThe group $(\\mathbb{Z}^n, +)$ is finitely generated by the $n$ standard basis vectors.\nThe 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.\n\n## Explanations\n\n### L1 human\n\nA group is finitely generated when a finite list of elements is enough to build all elements.\nThe group may still be infinite, as the integers show: one generator $1$ gives all integers.\nThe rational numbers need infinitely many generators.\n",
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