Math 2.0

Group theory and generalizations Infinite groups and combinatorial group theory Problem

Group isomorphism problem

Builds on Finitely presented groupPresentation of a groupGroup isomorphism

Problem

The group isomorphism problem asks for an algorithm that takes two finite presentations of groups and decides whether the groups they define are isomorphic. The problem may also be posed inside a class of finitely presented groups: the input is two presentations of groups of the class.

Remarks

The problem is unsolvable in general. By the Adian–Rabin theorem, there is no algorithm that decides whether a finite presentation defines the trivial group. Since the trivial group has the finite presentation ⟨ ∣ ⟩\langle \ \mid \ \rangle, no algorithm decides isomorphism of finitely presented groups.

The problem is solvable for some classes. For finitely generated abelian groups given by finite presentations, the Smith normal form of the relation matrix gives the invariant factors. Two such groups are isomorphic exactly when the invariants agree. It is also solvable for hyperbolic groups. It is one of the three decision problems, with the word problem and the conjugacy problem, that Max Dehn posed.

For 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.

Reading path

To understand this page from the ground up, read these 4 pages in order. The foundations come first.

Show the path as a graph
Finitely presented group, concept, no evidence levelFinitely presented gr…conceptGroup isomorphism, concept, no evidence levelGroup isomorphismconceptPresentation of a group, concept, no evidence levelPresentation of a gro…conceptGroup isomorphism problem, problem, no evidence levelGroup isomorphism pro…problem
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Finitely presented group concept
  2. Group isomorphism concept
  3. Presentation of a group concept
  4. Group isomorphism problem problem

Further reading

  1. Web https://en.wikipedia.org/wiki/Group_isomorphism_problem
About this page
TagMP3F49QT3C
URIm2://MP3F49QT3C
Kindproblem
WritingW0 stub
MSC20F10, 03D40
StatusThis copy of the site has no build data. Run m2 status MP3F49QT3C.
AboutGroup isomorphism, Finitely presented group
UsesFinitely presented group, Presentation of a group, Group isomorphism
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Changed2 November 2026

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