Schreier refinement theorem
E621109
The Schreier refinement theorem is a result in group theory stating that any two subnormal series of a group admit equivalent refinements, serving as a precursor and companion to the Jordan–Hölder theorem.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Schreier refinement theorem canonical | 2 |
How this entity was disambiguated
This entity first appeared as the object of triple T6833950 — 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: Schreier refinement theorem Context triple: [Jordan–Hölder theorem, relatedTo, Schreier refinement theorem]
-
A.
Jordan–Hölder theorem
The Jordan–Hölder theorem is a fundamental result in group theory stating that any two composition series of a finite group have the same length and the same (up to order and isomorphism) simple factor groups.
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B.
Noether's isomorphism theorems
Noether's isomorphism theorems are fundamental results in abstract algebra that relate quotient structures and substructures of groups, rings, and modules, providing a unifying framework for understanding homomorphic images and factor structures.
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C.
Sylow theorems
The Sylow theorems are fundamental results in finite group theory that describe the existence, conjugacy, and number of subgroups whose orders are powers of a prime dividing the group order.
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D.
Cauchy's theorem in group theory
Cauchy's theorem in group theory is a fundamental result stating that if a finite group’s order is divisible by a prime p, then the group contains an element (and hence a subgroup) of order p.
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E.
Artin–Schreier theory
Artin–Schreier theory is a branch of algebraic number theory and field theory that characterizes cyclic extensions of prime degree in fields of characteristic p using additive polynomials.
- 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: Schreier refinement theorem Target entity description: The Schreier refinement theorem is a result in group theory stating that any two subnormal series of a group admit equivalent refinements, serving as a precursor and companion to the Jordan–Hölder theorem.
-
A.
Jordan–Hölder theorem
The Jordan–Hölder theorem is a fundamental result in group theory stating that any two composition series of a finite group have the same length and the same (up to order and isomorphism) simple factor groups.
-
B.
Noether's isomorphism theorems
Noether's isomorphism theorems are fundamental results in abstract algebra that relate quotient structures and substructures of groups, rings, and modules, providing a unifying framework for understanding homomorphic images and factor structures.
-
C.
Sylow theorems
The Sylow theorems are fundamental results in finite group theory that describe the existence, conjugacy, and number of subgroups whose orders are powers of a prime dividing the group order.
-
D.
Cauchy's theorem in group theory
Cauchy's theorem in group theory is a fundamental result stating that if a finite group’s order is divisible by a prime p, then the group contains an element (and hence a subgroup) of order p.
-
E.
Artin–Schreier theory
Artin–Schreier theory is a branch of algebraic number theory and field theory that characterizes cyclic extensions of prime degree in fields of characteristic p using additive polynomials.
- F. None of above. chosen
Statements (29)
| Predicate | Object |
|---|---|
| instanceOf | theorem ⓘ |
| appliesTo |
finite groups
ⓘ
infinite groups ⓘ subnormal series of a group ⓘ |
| concerns |
composition series
ⓘ
factor groups ⓘ groups ⓘ refinement of series ⓘ subnormal series ⓘ |
| field | group theory ⓘ |
| hasConcept |
equivalent refinements
ⓘ
isomorphic factor groups ⓘ refinement of a series of subgroups ⓘ |
| hasConsequence | any two composition series of a group are equivalent (together with additional arguments) ⓘ |
| implies | any two subnormal series of a group have refinements with isomorphic factor groups up to order ⓘ |
| isCompanionOf | Jordan–Hölder theorem ⓘ |
| isPrecursorOf | Jordan–Hölder theorem ⓘ |
| namedAfter | Otto Schreier ⓘ |
| refines | Jordan–Hölder theorem ⓘ |
| requires |
definition of factor group
ⓘ
definition of normal subgroup ⓘ definition of subnormal subgroup ⓘ |
| statement | Any two subnormal series of a group admit equivalent refinements. ⓘ |
| topic |
equivalence of series
ⓘ
series of subgroups ⓘ structure of groups ⓘ |
| usedIn |
abstract algebra textbooks
ⓘ
classification of series of subgroups ⓘ proofs of Jordan–Hölder theorem ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Schreier refinement theorem Description of subject: The Schreier refinement theorem is a result in group theory stating that any two subnormal series of a group admit equivalent refinements, serving as a precursor and companion to the Jordan–Hölder theorem.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.