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  "body": "\n## Definition\n\nLet $G$ be a [[group|G5S1M9]].\nFor a set $S$ and a subset $R$ of the [[free group|0KTW26C80J]] $F(S)$, let $N(R)$ be the [[normal closure|6XM4CCVTKE]] of $R$ in $F(S)$.\nThe group $G$ is *finitely presented* if it is [[isomorphic|08MG7H8JW5]] to $\\langle S \\mid R \\rangle = F(S) / N(R)$ for some finite $S$ and finite $R$.\n\n## Remarks\n\nThe pair $(S, R)$ is a finite [[presentation|0JGEQXQ4AC]] of $G$.\nExample: $\\Z^2 \\cong \\langle a, b \\mid a b a^{-1} b^{-1} \\rangle$; the map $a \\mapsto (1, 0)$, $b \\mapsto (0, 1)$ induces the isomorphism, since the quotient is abelian and generated by two elements, and $\\Z^2$ is the free abelian group on them.\nEvery finite group $G$ is finitely presented, with $S = G$ and $R = \\set{ g h k^{-1} : g, h, k \\in G, \\ gh = k \\text{ in } G}$.\nNon-example: the additive group $\\Q$ is not finitely presented; it fails already because it is not [[finitely generated|082DWWXMRJ]], since a subgroup generated by finitely many fractions has bounded denominators.\nA finitely generated group need not be finitely presented; the [[lamplighter group|4GNPFMT7DZ]] $\\Z/2 \\wr \\Z$ is such an example.\nIf $G$ is finitely presented, then for every finite generating set $S$ of $G$ there is a finite set of relations $R$ with $G \\cong \\langle S \\mid R \\rangle$.\n",
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