Whitehead problem

In group theory, the Whitehead problem is the following question:

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

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. Abelian groups satisfying this condition are sometimes called Whitehead groups, so Whitehead's problem asks if every Whitehead group is free.

The question was asked by J. H. C. Whitehead in the 1950s, motivated by the second Cousin problem. The affirmative answer for countable groups was proved by Stein (1951). Progress for larger groups was slow, and the problem was considered an important one in algebra for many years.

Saharon Shelah (1974) showed that from the standard ZFC axiom system, the statement can be neither proven nor disproven. More precisely, he showed that:

• If every set is constructible, then every Whitehead group is free.
• If Martin's axiom holds and the continuum hypothesis is false, then there is a non-free Whitehead group.

Since the consistency of ZFC implies both the consistency of the axiom that all sets are constructible, and the consistency of Martin's axiom plus the negation of the continuum hypothesis, this shows that Whitehead's problem is undecidable.

This 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 that was shown to be undecidable.

The Whitehead problem remains undecidable even if one assumes the Continuum hypothesis, as shown by Saharon Shelah (1977, 1980). Various similar independence statements were proved, showing that the theory of uncountable abelian 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

• List of statements undecidable in ZFC
• Statements true in L