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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|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|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|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 [[#References|[a3]]] for results prior to 1973).

In 1973, S. Shelah [[#References|[a5]]] proved that it is undecidable under the axioms of Zermelo–Frankel set theory (cf. also [[Set theory|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|Gödel constructive set]]), but it is false assuming Martin's axiom (cf. [[Suslin hypothesis|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 [[#References|[a6]]]. He also proved that the problem is undecidable even if  $  { \mathop{\rm CH} } $
is assumed [[#References|[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 [[#References|[a8]]].

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

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