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  "body": "\n## Problem\n\nThe *group isomorphism problem* asks for an algorithm that takes two finite [[presentations|0JGEQXQ4AC]] of groups and decides whether the groups they define are [[isomorphic|08MG7H8JW5]].\nThe problem may also be posed inside a class of [[finitely presented groups|TJKANTKASZ]]: the input is two presentations of groups of the class.\n\n## Remarks\n\nThe problem is unsolvable in general.\nBy the Adian–Rabin theorem, there is no algorithm that decides whether a finite presentation defines the trivial group.\nSince the trivial group has the finite presentation $\\langle \\ \\mid \\ \\rangle$, no algorithm decides isomorphism of finitely presented groups.\n\nThe problem is solvable for some classes.\nFor finitely generated abelian groups given by finite presentations, the Smith normal form of the relation matrix gives the invariant factors.\nTwo such groups are isomorphic exactly when the invariants agree.\nIt is also solvable for [[hyperbolic groups|JWJ8YPR7EB]].\nIt is one of the three decision problems, with the [[word problem|PBGTTAVMMT]] and the [[conjugacy problem|4AF7AF7690]], that Max Dehn posed.\n\nFor finite groups given by their multiplication tables the question is always decidable, and the interest is in the running time; this is a different, computational problem.\n\n## Explanations\n\n### L1 human\n\nThe same group can be written with very different generators and rules.\nFor example, $\\langle a, b \\mid a^2, b^3, a b a^{-1} b^{-1} \\rangle$ and $\\langle c \\mid c^6 \\rangle$ both define the cyclic group of order $6$, with $c = ab$.\nThe isomorphism problem asks whether a computer can always tell when two descriptions define the same group.\nIn general it cannot: no program can even decide whether a given description defines the group with one element.\n",
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      "first_stated": "1911",
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