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A problem attributed, to J.H.C. Whitehead, which asks for a characterization of Abelian groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110030/w1100301.png" /> that satisfy the homological condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110030/w1100302.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110030/w1100303.png" /> 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: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110030/w1100304.png" /> is a Whitehead group if and only if for every surjective homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110030/w1100305.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110030/w1100306.png" />, if the kernel of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110030/w1100307.png" /> is isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110030/w1100308.png" />, then the kernel is a direct summand of the domain of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110030/w1100309.png" />. A sufficient condition for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110030/w11003010.png" /> to be a Whitehead group is that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110030/w11003011.png" /> is free (see [[Free Abelian group|Free Abelian group]]). This condition has been proved to be necessary if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110030/w11003012.png" /> is countable. Since a subgroup of a Whitehead group is again a Whitehead group, this means that every Whitehead group is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110030/w11003014.png" />-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]] (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110030/w11003015.png" />), they were proved to be strongly <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110030/w11003017.png" />-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).
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110030/w11003018.png" /> is free. Specifically, he proved that this is true assuming the Gödel constructibility axiom, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110030/w11003019.png" /> (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 (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110030/w11003020.png" />). Later he proved that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110030/w11003021.png" /> implies that every Whitehead group, of arbitrary cardinality, is free [[#References|[a6]]]. He also proved that the problem is undecidable even if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110030/w11003022.png" /> is assumed [[#References|[a7]]] and that it is consistent with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110030/w11003023.png" /> that there are Whitehead groups of cardinality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110030/w11003024.png" /> that are not strongly <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110030/w11003025.png" />-free [[#References|[a8]]].
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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]]].
 
For more information see [[#References|[a1]]], [[#References|[a4]]] or [[#References|[a2]]].

Latest revision as of 08:29, 6 June 2020


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
How to Cite This Entry:
Whitehead problem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Whitehead_problem&oldid=17939
This article was adapted from an original article by P.C. Eklof (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article