Dehn’s decision problems in group theory
E265414
Dehn’s decision problems in group theory are foundational early 20th-century problems that introduced algorithmic questions about the solvability of word, conjugacy, and isomorphism problems in finitely presented groups, helping launch the field of algorithmic group theory.
All labels observed (3)
| Label | Occurrences |
|---|---|
| Dehn’s conjugacy problem | 1 |
| Dehn’s decision problems | 1 |
| Dehn’s decision problems in group theory canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T2416871 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Dehn’s decision problems in group theory Context triple: [Max Dehn, notableWork, Dehn’s decision problems in group theory]
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A.
Knuth–Bendix completion algorithm
The Knuth–Bendix completion algorithm is a procedure in term rewriting and automated theorem proving that transforms a set of equations into a confluent rewriting system, enabling decision of word problems in algebraic structures.
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B.
The Poincaré-Birkhoff-Witt theorem in ring theory
"The Poincaré-Birkhoff-Witt theorem in ring theory" is a mathematical work, attributed here to N. G. de Bruijn, that studies and applies the Poincaré–Birkhoff–Witt theorem in the context of associative and Lie-theoretic ring structures.
-
C.
Hilbert’s tenth problem
Hilbert’s tenth problem is a famous unsolved question in mathematics that asked for a general algorithm to determine whether any given Diophantine equation has an integer solution, and whose negative answer helped establish fundamental limits of computability.
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D.
Entscheidungsproblem
The Entscheidungsproblem is a foundational decision problem in mathematical logic that asks whether there exists a general algorithm to determine the truth or falsity of any given first-order logical statement.
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E.
Hilbert’s fourteenth problem
Hilbert’s fourteenth problem is one of David Hilbert’s famous list of 23 problems, concerning the finite generation of certain algebras of invariants in algebraic geometry and invariant theory.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Dehn’s decision problems in group theory Target entity description: Dehn’s decision problems in group theory are foundational early 20th-century problems that introduced algorithmic questions about the solvability of word, conjugacy, and isomorphism problems in finitely presented groups, helping launch the field of algorithmic group theory.
-
A.
Knuth–Bendix completion algorithm
The Knuth–Bendix completion algorithm is a procedure in term rewriting and automated theorem proving that transforms a set of equations into a confluent rewriting system, enabling decision of word problems in algebraic structures.
-
B.
The Poincaré-Birkhoff-Witt theorem in ring theory
"The Poincaré-Birkhoff-Witt theorem in ring theory" is a mathematical work, attributed here to N. G. de Bruijn, that studies and applies the Poincaré–Birkhoff–Witt theorem in the context of associative and Lie-theoretic ring structures.
-
C.
Hilbert’s tenth problem
Hilbert’s tenth problem is a famous unsolved question in mathematics that asked for a general algorithm to determine whether any given Diophantine equation has an integer solution, and whose negative answer helped establish fundamental limits of computability.
-
D.
Entscheidungsproblem
The Entscheidungsproblem is a foundational decision problem in mathematical logic that asks whether there exists a general algorithm to determine the truth or falsity of any given first-order logical statement.
-
E.
Hilbert’s fourteenth problem
Hilbert’s fourteenth problem is one of David Hilbert’s famous list of 23 problems, concerning the finite generation of certain algebras of invariants in algebraic geometry and invariant theory.
- F. None of above. chosen
Statements (44)
| Predicate | Object |
|---|---|
| instanceOf |
collection of decision problems in group theory
ⓘ
mathematical concept ⓘ |
| aim | determine whether certain questions about group elements are algorithmically decidable ⓘ |
| appliesTo | finitely presented groups ⓘ |
| concerns |
computational aspects of group presentations
ⓘ
existence of decision procedures ⓘ solvability of algorithmic problems in groups ⓘ |
| context |
combinatorial group theory
ⓘ
mathematical logic ⓘ |
| field |
algorithmic group theory
ⓘ
computability theory ⓘ group theory ⓘ |
| formulatedBy | Max Dehn ⓘ |
| hasImpactOn |
classification of groups by algorithmic properties
ⓘ
complexity theory in group theory ⓘ study of recursively presented groups ⓘ |
| hasPart |
Dehn’s conjugacy problem
ⓘ
linked to:
Dehn’s decision problems in group theory
Dehn’s isomorphism problem ⓘ Dehn’s word problem ⓘ
linked to:
Dehn algorithm
|
| historicalSignificance |
helped launch algorithmic group theory
ⓘ
introduced algorithmic questions into group theory ⓘ |
| inception | early 20th century ⓘ |
| influenced |
development of undecidability results in group theory
ⓘ
study of algorithmic properties of finitely presented groups ⓘ |
| inspiredBy | combinatorial methods in topology ⓘ |
| mainSubject |
conjugacy problem for groups
ⓘ
isomorphism problem for groups ⓘ word problem for groups ⓘ |
| motivated |
development of normal form algorithms in groups
ⓘ
search for classes of groups with solvable word problem ⓘ study of decision problems in other algebraic structures ⓘ |
| namedAfter | Max Dehn ⓘ |
| notableResult |
the isomorphism problem is unsolvable for general finitely presented groups
ⓘ
there exist finitely presented groups with unsolvable conjugacy problem ⓘ there exist finitely presented groups with unsolvable word problem ⓘ |
| relatedTo |
computational group theory
ⓘ
decision problems in algebra ⓘ word problem for semigroups ⓘ |
| status | some instances are decidable and some are undecidable ⓘ |
| typicalQuestion |
Is there an algorithm to decide whether a word represents the identity in a given finitely presented group?
ⓘ
Is there an algorithm to decide whether two elements of a group are conjugate? ⓘ Is there an algorithm to decide whether two finitely presented groups are isomorphic? ⓘ |
| usedIn |
analysis of automatic groups
ⓘ
analysis of hyperbolic groups ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Dehn’s decision problems in group theory Description of subject: Dehn’s decision problems in group theory are foundational early 20th-century problems that introduced algorithmic questions about the solvability of word, conjugacy, and isomorphism problems in finitely presented groups, helping launch the field of algorithmic group theory.
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Dehn’s decision problems in group theory
linked to: Dehn’s decision problems in group theory