Whitehead problem

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

Is every abelian group A with Ext[?]¹(A, Z) = 0 a free abelian group?

This was asked by J. H. C. Whitehead[?] in the 1950s, motivated by the second Cousin problem[?]. The affirmative answer for countable groups was already found in the 1950s. Progress for larger groups was slow, and the problem was considered one of the most important ones in algebra for many years.

In 1973, Saharon Shelah[?] showed that from the standard ZFC axiom system, the statement can be neither proven nor disproven.

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 is undecidable even if one assumes the Continuum hypothesis, as shown by Shelah in 1980.

References

• S. Shelah: "Infinite Abelian groups, Whitehead problem and some constructions", Israel Journal of Mathematics 18 (1974), pp. 243-256.
• S. Shelah: "Whitehead groups may not be free, even assuming CH. II", Israel Journal of Mathematics 35 (1980), pp. 257-285