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  "body": "\n## Summary\n\nA finitely presented group is a group that has a [[presentation|0JGEQXQ4AC]] with finitely many generators and finitely many relations.\nSuch a group can be described by a finite amount of data, so it can be handled by algorithms and computers.\nFinitely presented groups are the natural objects of combinatorial group theory and arise as [[fundamental groups|S1XK3HXHFX]] of compact manifolds and finite complexes.\n\n## Explanations\n\n### L1 human\n\nA finitely presented group is given by a finite list of letters and a finite list of rules that say which words equal the identity.\nFor example, $\\langle a, b \\mid a b a^{-1} b^{-1} \\rangle$ has two letters and one rule, $ab = ba$, and it is the group $\\Z^2$.\nEvery finite group is finitely presented, since its whole multiplication table is a finite list of rules.\nSome finitely generated groups need infinitely many rules, however one chooses the generators.\n",
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