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?
Refinement
Assume that A is an abelian group such that every short exact sequence
must split if B is also abelian. The Whitehead problem then asks: must A be free? This splitting requirement is equivalent to the condition Ext1(A, Z) = 0. Abelian groups A satisfying this condition are sometimes called Whitehead groups, so Whitehead's problem asks: is every Whitehead group free? It should be mentioned that if this condition is strengthened by requiring that the exact sequence
must split for any abelian group C, then it is well known that this is equivalent to A being 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
Saharon Shelah showed that, given the canonical
- 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 both 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 cannot be resolved in ZFC.
Discussion
Shelah's result was completely unexpected. While the existence of undecidable statements had been known since
Shelah later showed that the Whitehead problem remains undecidable even if one assumes the continuum hypothesis.
See also
- Free abelian group
- Whitehead torsion
- List of statements undecidable in ZFC
- Statements true in L
References
Further reading
- Eklof, Paul C. (December 1976). "Whitehead's Problem is Undecidable". The American Mathematical Monthly. 83 (10): 775–788. JSTOR 2318684. An expository account of Shelah's proof.
- Eklof, P.C. (2001) [1994], "Whitehead problem", Encyclopedia of Mathematics, EMS Press