Math 2.0

Group theory and generalizations Abelian groups Problem

Whitehead problem

Builds on Abelian groupGroup homomorphismFree abelian groupExt functorZermelo–Fraenkel set theory

Problem

Let AA be an abelian group. Then AA is a Whitehead group if the following holds for every abelian group BB. For every surjective homomorphism f ⁣:B→Af \colon B \to A with kernel isomorphic to ℤ\Z, there is a homomorphism g ⁣:A→Bg \colon A \to B with f∘g=id⁡Af \circ g = \id_A. Equivalently, Ext1(A,ℤ)=0\mathrm{Ext}^1(A, \Z) = 0, in terms of the Ext functor. The Whitehead problem asks: is every Whitehead group a free abelian group?

Remarks

Every free abelian group is a Whitehead group, because a surjection onto a free abelian group always splits: send each basis element to any preimage. Every countable Whitehead group is free, so the question concerns uncountable groups. Shelah showed in 1974 that the problem is independent of ZFC. If the axiom of constructibility V=LV = L holds, then every Whitehead group is free. If Martin's axiom holds together with the negation of the continuum hypothesis, then there is a Whitehead group that is not free.

Reading path

To understand this page from the ground up, read these 19 pages in order. The foundations come first.

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Abelian group, concept, no evidence levelAbelian groupconceptElement, definition, no evidence levelElementdefinitionExt functor, concept, no evidence levelExt functorconceptFree abelian group, concept, no evidence levelFree abelian groupconceptGroup homomorphism, concept, no evidence levelGroup homomorphismconceptProposition, concept, no evidence levelPropositionconceptSet, definition, no evidence levelSetdefinitionConjunction, definition, no evidence levelConjunctiondefinitionDisjunction, definition, no evidence levelDisjunctiondefinitionImplication, definition, no evidence levelImplicationdefinitionNegation, definition, no evidence levelNegationdefinitionPredicate, definition, no evidence levelPredicatedefinitionBiconditional, definition, no evidence levelBiconditionaldefinitionUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionAxiom of extensionality, statement, no evidence levelAxiom of extensionali…statementPropositional logic, concept, no evidence levelPropositional logicconceptPredicate logic, concept, no evidence levelPredicate logicconceptZermelo–Fraenkel set theory, concept, no evidence levelZermelo–Fraenkel set …conceptWhitehead problem, problem, no evidence levelWhitehead problemproblem
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. Abelian group concept
  2. Element definition
  3. Ext functor concept
  4. Free abelian group concept
  5. Group homomorphism concept
  6. Proposition concept
  7. Set definition
  8. Conjunction definition
  9. Disjunction definition
  10. Implication definition
  11. Negation definition
  12. Predicate definition
  13. Biconditional definition
  14. Universal quantifier definition
  15. Axiom of extensionality statement
  16. Propositional logic concept
  17. Predicate logic concept
  18. Zermelo–Fraenkel set theory concept
  19. Whitehead problem problem

Further reading

  1. Web https://en.wikipedia.org/wiki/Whitehead_problem
About this page
TagF0B6BRJA5Z
URIm2://F0B6BRJA5Z
Kindproblem
WritingW0 stub
MSC20K20, 20K35, 03E35
StatusThis copy of the site has no build data. Run m2 status F0B6BRJA5Z.
AboutFree abelian group, Abelian group
UsesAbelian group, Group homomorphism, Free abelian group, Ext functor, Zermelo–Fraenkel set theory
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Changed2 November 2026

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