Whitehead problem

A problem attributed, to J.H.C. Whitehead, which asks for a characterization of Abelian groups $ A $ that satisfy the homological condition $ { \mathop{\rm Ext} } ( A, \mathbf Z ) = 0 $, where $ \mathbf Z $ is the group of integers under addition (cf. also Homology). A group which satisfies this condition is called a Whitehead group. An equivalent characterization is: $ A $ is a Whitehead group if and only if for every surjective homomorphism $ \pi $ onto $ A $, if the kernel of $ \pi $ is isomorphic to $ \mathbf Z $, then the kernel is a direct summand of the domain of $ \pi $. A sufficient condition for $ A $ to be a Whitehead group is that $ A $ is free (see Free Abelian group). This condition has been proved to be necessary if $ A $ is countable. Since a subgroup of a Whitehead group is again a Whitehead group, this means that every Whitehead group is $ \aleph _ {1} $- free, that is, every countable subgroup is free. Before 1973 only partial results were obtained for uncountable groups: Whitehead groups were proved to be separable and slender, and under the assumption of the continuum hypothesis ( $ { \mathop{\rm CH} } $), they were proved to be strongly $ \aleph _ {1} $- free, that is, every countable subset is contained in a countable free subgroup which is a direct summand of countable subgroups containing it (see [a3] for results prior to 1973).

In 1973, S. Shelah [a5] proved that it is undecidable under the axioms of Zermelo–Frankel set theory (cf. also Set theory), ZFC, whether every Whitehead group of cardinality $ \aleph _ {1} $ is free. Specifically, he proved that this is true assuming the Gödel constructibility axiom, $ V = L $( see Gödel constructive set), but it is false assuming Martin's axiom (cf. Suslin hypothesis) and the negation of the continuum hypothesis ( $ { \mathop{\rm MA} } + \neg { \mathop{\rm CH} } $). Later he proved that $ V = L $ implies that every Whitehead group, of arbitrary cardinality, is free [a6]. He also proved that the problem is undecidable even if $ { \mathop{\rm CH} } $ is assumed [a7] and that it is consistent with $ \neg { \mathop{\rm CH} } $ that there are Whitehead groups of cardinality $ \aleph _ {1} $ that are not strongly $ \aleph _ {1} $- free [a8].

For more information see [a1], [a4] or [a2].

References

[a1]	P.C. Eklof, "Whitehead's problem is undecidable" Amer. Math. Monthly , 83 (1976) pp. 775–788
[a2]	P.C. Eklof, A.H. Mekler, "Almost free modules" , North-Holland (1990)
[a3]	L. Fuchs, "Infinite Abelian groups" , 2 , Acad. Press (1973)
[a4]	R. J. Nunke, "Whitehead's problem" , Abelian Group Theory , Lecture Notes in Mathematics , 616 , Springer (1977) pp. 240–250
[a5]	S. Shelah, "Infinite abelian groups, Whitehead problem and some constructions" Israel J. Math. , 18 (1974) pp. 243–25
[a6]	S. Shelah, "A compactness theorem for singular cardinals, free algebras, Whitehead problem and transversals" Israel J. Math. , 21 (1975) pp. 319–349
[a7]	S. Shelah, "Whitehead groups may not be free even assuming CH, I" Israel J. Math. , 28 (1977) pp. 193–203
[a8]	S. Shelah, "On uncountable abelian groups" Israel J. Math. , 32 (1979) pp. 311–330
[a9]	S. Shelah, "Whitehead groups may not be free even assuming CH, II" Israel J. Math. , 35 (1980) pp. 257–285