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Whitehead problem
In group theory, a branch of abstract algebra, the Whitehead problem is the following question:
Is every abelian group A with Ext1(A, Z) = 0 a free abelian group?
Saharon Shelah proved that Whitehead's problem is independent of ZFC, the standard axioms of set theory.
Assume that A is an abelian group such that every short exact sequence
must split if B is also abelian. The Whitehead problem then asks: must A be free? This splitting requirement is equivalent to the condition Ext1(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
must split for any abelian group C, then it is known that this is equivalent to A being free. (See Projective module).
Caution: The converse of Whitehead's problem, namely that every free abelian group is Whitehead, is a known group-theoretical fact. Some authors call Whitehead group only a non-free group A satisfying Ext1(A, Z) = 0. Whitehead's problem then asks: do Whitehead groups exist?
Saharon Shelah showed that, given the canonical ZFC axiom system, the problem is independent of the usual axioms of set theory. More precisely, he showed that:
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Whitehead problem
In group theory, a branch of abstract algebra, the Whitehead problem is the following question:
Is every abelian group A with Ext1(A, Z) = 0 a free abelian group?
Saharon Shelah proved that Whitehead's problem is independent of ZFC, the standard axioms of set theory.
Assume that A is an abelian group such that every short exact sequence
must split if B is also abelian. The Whitehead problem then asks: must A be free? This splitting requirement is equivalent to the condition Ext1(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
must split for any abelian group C, then it is known that this is equivalent to A being free. (See Projective module).
Caution: The converse of Whitehead's problem, namely that every free abelian group is Whitehead, is a known group-theoretical fact. Some authors call Whitehead group only a non-free group A satisfying Ext1(A, Z) = 0. Whitehead's problem then asks: do Whitehead groups exist?
Saharon Shelah showed that, given the canonical ZFC axiom system, the problem is independent of the usual axioms of set theory. More precisely, he showed that: