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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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Latest revision as of 04:18, 25 December 2014
Hi there, I am Andrew Berryhill. Kentucky is where I've usually been living. To climb is something she would by no means give up. Office supervising is where her primary income comes from but she's already utilized for another 1.
Here is my web-site; love psychics