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?

Abelian groups satisfying this condition are sometimes called Whitehead groups, so Whitehead's problem asks: is every Whitehead group free? Shelah (1974) proved that Whitehead's problem was undecidable within standard ZFC set theory.

Refinement

The condition Ext¹(A, Z) = 0 can be equivalently formulated as follows: whenever B is an abelian group and f : B → A is a surjective group homomorphism whose kernel is isomorphic to the group of integers Z, then there exists a group homomorphism g : A → B with fg = id_A.

Shelah's proof

Saharon Shelah (1974) showed that, given the canonical ZFC axiom system, the problem was undecidable. More precisely, he showed that:

• If every set is constructible, then every Whitehead group is free;
• If Martin's axiom and the negation of the continuum hypothesis both hold, then there is a non-free Whitehead group.

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

• The axiom of constructibility (which asserts that all sets are constructible);
• Martin's axiom plus the negation of the continuum hypothesis,
  

Whitehead's problem is undecidable.

Discussion

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 been confined to the realm of set theory. The Whitehead problem was the first purely algebraic problem to be proved undecidable.

Shelah (1977, 1980) later showed that the Whitehead problem remains undecidable even if one assumes the Continuum hypothesis. Proving that this and other statements about uncountable abelian groups are independent of ZFC shows that the theory of such groups depends very sensitively on the underlying set theory.

References

• Eklof, Paul C. (1976), "Whitehead's Problem is Undecidable", The American Mathematical Monthly, 83 (10): 775–788 An expository account of Shelah's proof.
• Eklof, P.C. (2001) [1994], "Whitehead problem", Encyclopedia of Mathematics, EMS Press
• Shelah, S. (1974), "Infinite Abelian groups, Whitehead problem and some constructions", Israel Journal of Mathematics, 18: 243–256, MR0357114
• Shelah, S. (1977), "Whitehead groups may not be free, even assuming CH. I", Israel Journal of Mathematics, 28: 193–203, MR0469757
• Shelah, S. (1980), "Whitehead groups may not be free, even assuming CH. II", Israel Journal of Mathematics, 35: 257–285, MR0594332
• Stein, Karl (1951), "Analytische Funktionen mehrerer komplexer Veränderlichen zu vorgegebenen Periodizitätsmoduln und das zweite Cousinsche Problem", Math. Ann., 123: 201–222, doi:10.1007/BF02054949, MR0043219

See also

• Free abelian group
• Whitehead group
• List of statements undecidable in ZFC
• Statements true if all sets are constructible