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In [[mathematics]], in the realm of [[abelian group|abelian]] [[group theory]], an abelian [[group (mathematics)|group]] is said to be '''cotorsion''' if every extension of it by a [[torsion-free group]] splits. If the group is <math>C</math>, this is equivalent to asserting that <math>Ext(G,C) = 0</math> for all torsion-free groups <math>G</math>. It suffices to check the condition for <math>G</math> being the group of [[rational number]]s. | |||
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 group|algebraically compact]]. | |||
* [[Ulm subgroup]]s of cotorsion groups are cotorsion and [[Ulm factor]]s of cotorsion groups are algebraically compact. | |||
==External links== | |||
*{{SpringerEOM| title=Cotorsion group | id=Cotorsion_group | oldid=18282 | first=L. | last=Fuchs }} | |||
[[Category:Abelian group theory]] | |||
[[Category:Properties of groups]] | |||
Revision as of 22:26, 30 December 2013
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 , this is equivalent to asserting that for all torsion-free groups . It suffices to check the condition for 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
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