Editing Whitehead problem

{{Distinguish|Whitehead theorem|Whitehead conjecture}}
{{Use shortened footnotes|date=May 2021}}  
In [[group theory]], a branch of [[abstract algebra]], the '''Whitehead problem''' is the following question:

{{quote|1=Is every [[abelian group]] ''A'' with [[Ext functor|Ext]]<sup>1</sup>(''A'', '''Z''') = 0 a [[free abelian group]]?}}

[[Saharon Shelah]] proved that Whitehead's problem is [[Independence (mathematical logic)|independent]] of [[ZFC]], the standard axioms of set theory.{{r|Shelah1974}}

==Refinement==
Assume that ''A'' is an abelian group such that every short [[exact sequence]]
:<math>0\rightarrow\mathbb{Z}\rightarrow B\rightarrow A\rightarrow 0</math>
must [[Split_exact_sequence|split]] if ''B'' is also abelian. The Whitehead problem then asks: must ''A'' be free? This splitting requirement is equivalent to the condition Ext<sup>1</sup>(''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 
:<math>0\rightarrow C\rightarrow B\rightarrow A\rightarrow 0</math>
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 Ext<sup>1</sup>(''A'', '''Z''') = 0. Whitehead's problem then asks: do Whitehead groups exist?

==Shelah's proof==
Saharon Shelah showed that,  given the canonical [[ZFC]] axiom system, the problem is [[Independence (mathematical logic)|independent of the usual axioms of set theory]].{{r|Shelah1974}} More precisely, he showed that:
* If [[Axiom of constructibility|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==
[[J. H. C. Whitehead]], motivated by the [[second Cousin problem]], first posed the problem in the 1950s. Stein answered the question in the affirmative for [[countable]] groups.{{r|Stein1951}} Progress for larger groups was slow, and the problem was considered an important one in [[abstract algebra|algebra]] for some years.

Shelah's result was completely unexpected. While the existence of undecidable statements had been known since [[Gödel's incompleteness theorem]] of 1931, previous examples of undecidable statements (such as the [[continuum hypothesis]]) had all been in pure [[set theory]]. The Whitehead problem was the first purely algebraic problem to be proved undecidable.

Shelah later showed that the Whitehead problem remains undecidable even if one assumes the continuum hypothesis.{{r|Shelah1977|Shelah1980}} In fact, it remains undecidable even under the [[generalised continuum hypothesis]].<ref>{{cite web |url=https://www.karlin.mff.cuni.cz/~trlifaj/ANK_5.pdf |title=The Whitehead Problem and Beyond (Lecture notes for NMAG565) |last=Triflaj |first=Jan |date=16 February 2023 |website= |publisher=[[Charles University]] |access-date=26 September 2024 |quote=}}</ref> The Whitehead conjecture is true if all sets are [[constructible universe|constructible]]. That this and other statements about uncountable abelian groups are provably independent of [[ZFC]] shows that the theory of such groups is very sensitive to the assumed underlying [[set theory]].

==See also==
*[[Free abelian group]]
*[[Whitehead torsion]]
*[[List of statements undecidable in ZFC]]
*[[Axiom of constructibility#Statements true in L|Statements true in L]]

== References ==
{{reflist|refs=
<ref name=Shelah1974>{{cite journal |last=Shelah |first=S. |author-link=Saharon Shelah |date=1974 |title=Infinite Abelian groups, Whitehead problem and some constructions |journal=[[Israel Journal of Mathematics]] |volume=18 |issue=3 |pages=243–256 |doi=10.1007/BF02757281 | doi-access= |mr=0357114 |s2cid=123351674}}</ref>

<ref name=Stein1951>{{cite journal |last=Stein |first=Karl |date=1951 |title=Analytische Funktionen mehrerer komplexer Veränderlichen zu vorgegebenen Periodizitätsmoduln und das zweite Cousinsche Problem |journal= Mathematische Annalen |volume=123 |pages=201–222 |doi=10.1007/BF02054949 | doi-access= |mr=0043219|s2cid= 122647212 }}</ref>

<ref name=Shelah1977>{{cite journal |last=Shelah |first=S. |author-link=Saharon Shelah |date=1977 |title=Whitehead groups may not be free, even assuming CH. I
|journal=[[Israel Journal of Mathematics]] |volume=28 |issue=3 |page=193-203 |doi=10.1007/BF02759809 | doi-access=free |mr=0469757 |hdl=10338.dmlcz/102427 |hdl-access=free |s2cid=123029484}}</ref>

<ref name=Shelah1980>{{cite journal |last=Shelah |first=S. |author-link=Saharon Shelah |date=1980 |title=Whitehead groups may not be free, even assuming CH. II
|journal=[[Israel Journal of Mathematics]] |volume=35 |issue=4 |pages=257–285 |doi=10.1007/BF02760652 | doi-access= |mr=0594332 |s2cid=122336538}}</ref>
}}

==Further reading==
{{refbegin}}
*{{cite journal |last=Eklof |first=Paul C. |date=December 1976 |title=Whitehead's Problem is Undecidable |journal=The American Mathematical Monthly |volume= 83|issue= 10 |pages=775–788 |doi= 10.2307/2318684 |jstor=2318684}} An expository account of Shelah's proof.
*{{springer|id=W/w110030|title=Whitehead problem |author=Eklof, P.C.}}
{{refend}}

[[Category:Independence results]]
[[Category:Group theory]]
[[Category:Mathematical problems]]