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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.
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| Some properties of cotorsion groups:
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| * Any [[quotient]] of a cotorsion group is cotorsion.
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| * A [[direct product of groups]] is cotorsion [[if and only if]] each factor is.
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| * Every [[divisible group]] or [[injective group]] is cotorsion.
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| * 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.
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| * A torsion-free abelian group is cotorsion if and only if it is [[algebraically compact group|algebraically compact]].
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| * [[Ulm subgroup]]s of cotorsion groups are cotorsion and [[Ulm factor]]s of cotorsion groups are algebraically compact.
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| ==External links==
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| *{{SpringerEOM| title=Cotorsion group | id=Cotorsion_group | oldid=18282 | first=L. | last=Fuchs }}
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| [[Category:Abelian group theory]]
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| [[Category:Properties of groups]]
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Revision as of 12:30, 4 March 2014
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