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  "body": "\n## Definition\n\nLet $S$ be a set, let $F_S$ be the [[free group|0KTW26C80J]] on $S$, and let $R \\subseteq F_S$.\nLet $N$ be the smallest [[normal subgroup|0CB0DX5BVV]] of $F_S$ that contains $R$, the normal closure of $R$.\nThe group with presentation $\\langle S \\mid R \\rangle$ is the [[quotient group|0KYX2N69J0]] $F_S / N$.\nA group $G$ has the presentation $\\langle S \\mid R \\rangle$ when $G$ is [[isomorphic|08MG7H8JW5]] to $F_S / N$.\n\n## Remarks\n\nThe elements of $S$ are the generators and the elements of $R$ are the relators; a relation $x = y$ stands for the relator $y^{-1} x$.\nThe cyclic group of order $n$ has the presentation $\\langle a \\mid a^n \\rangle$, and the dihedral group of order $16$ has the presentation $\\langle r, f \\mid r^8, f^2, (r f)^2 \\rangle$.\nEach group has a presentation, and each finite group has a presentation with $S$ and $R$ finite.\nA group is finitely presented when it has a presentation with $S$ and $R$ finite.\n\n## Explanations\n\n### L1 human\n\nA presentation lists generators and the rules that they obey.\nThe group is the largest group that these generators make, subject only to these rules.\nPresentations are often the shortest way to describe a group.\n",
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