World Library  
Flag as Inappropriate
Email this Article

Whitehead problem

Article Id: WHEBN0000212619
Reproduction Date:

Title: Whitehead problem  
Author: World Heritage Encyclopedia
Language: English
Subject: Undecidable problem, Set-theoretic topology, Gödel's incompleteness theorems, List of group theory topics, Abelian group
Collection: Group Theory, Independence Results, Mathematical Problems
Publisher: World Heritage Encyclopedia

Whitehead problem

In group theory, a branch of abstract algebra, the Whitehead problem is the following question:

Is every abelian group A with Ext1(A, Z) = 0 a free abelian group?

Shelah (1974) proved that Whitehead's problem is undecidable within standard ZFC set theory.


  • Refinement 1
  • Shelah's proof 2
  • Discussion 3
  • See also 4
  • References 5


The condition Ext1(A, Z) = 0 can be equivalently formulated as follows: whenever B is an abelian group and f : BA is a surjective group homomorphism whose kernel is isomorphic to the group of integers Z, then there exists a group homomorphism g : AB with fg = idA. Abelian groups A satisfying this condition are sometimes called Whitehead groups, so Whitehead's problem asks: is every Whitehead group free?

Caution: The converse of Whitehead's problem, namely that every free abelian group is Whitehead, is a well known group-theoretical fact. Some authors call Whitehead group only a non-free group A satisfying Ext1(A, Z) = 0. Whitehead's problem then asks: do Whitehead groups exist?

Shelah's proof

showed that,  given the canonical ZFC axiom system, the problem is independent of the usual axioms of set theory. More precisely, he showed that:

Since the consistency of ZFC implies the consistency of either of the following:

Whitehead's problem cannot be resolved in ZFC.


J. H. C. Whitehead, motivated by the second Cousin problem, first posed the problem in the 1950s. Stein (1951) answered the question in the affirmative for countable groups. Progress for larger groups was slow, and the problem was considered an important one in algebra for some years.

Shelah's result was completely unexpected. While the existence of undecidable statements had been known since Gödel's incompleteness theorem of 1931, previous examples of undecidable statements (such as the continuum hypothesis) had all been in pure set theory. The Whitehead problem was the first purely algebraic problem to be proved undecidable.

later showed that the Whitehead problem remains undecidable even if one assumes the Continuum hypothesis. The Whitehead conjecture is true if all sets are constructible. That this and other statements about uncountable abelian groups are provably independent of ZFC shows that the theory of such groups is very sensitive to the assumed underlying set theory.

See also


  • An expository account of Shelah's proof.
This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.

Copyright © World Library Foundation. All rights reserved. eBooks from Project Gutenberg are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.