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# Cotorsion group

In mathematics, in the realm of abelian group theory, an abelian group is said to be cotorsion if every extension of it by a torsion-free group splits. If the group is C, this is equivalent to asserting that Ext(G,C) = 0 for all torsion-free groups G. It suffices to check the condition for G being the group of rational numbers.

Some properties of cotorsion groups:

Any quotient of a cotorsion group is cotorsion.

A direct product of groups is cotorsion if and only if each factor is.

Every divisible group or injective group is cotorsion.

The Baer Fomin Theorem states that a torsion group is cotorsion if and only if it is a direct sum of a divisible group and a bounded group, that is, a group of bounded exponent.

A torsion-free abelian group is cotorsion if and only if it is algebraically compact.

Ulm subgroups of cotorsion groups are cotorsion and Ulm factors of cotorsion groups are algebraically compact.

External links

Fuchs, L. (2001), "Cotorsion group", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

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