Whitehead problem
A problem attributed, to J.H.C. Whitehead, which asks for a characterization of Abelian groups
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 |
Whitehead problem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Whitehead_problem&oldid=49210