- 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.
Wikimedia Foundation. 2010.
Look at other dictionaries:
Algebraically compact group — In mathematics, in the realm of Abelian group theory, a group is said to be algebraically compact if it is a direct summand of every Abelian group containing it as a pure subgroup.Equivalent characterizations of algebraic compactness: * The group … Wikipedia
Abelian group — For other uses, see Abelian (disambiguation). Abelian group is also an archaic name for the symplectic group Concepts in group theory category of groups subgroups, normal subgroups group homomorphisms, kernel, image, quotient direct product,… … Wikipedia
List of mathematics articles (C) — NOTOC C C closed subgroup C minimal theory C normal subgroup C number C semiring C space C symmetry C* algebra C0 semigroup CA group Cabal (set theory) Cabibbo Kobayashi Maskawa matrix Cabinet projection Cable knot Cabri Geometry Cabtaxi number… … Wikipedia
Eilenberg-Moore spectral sequence — In mathematics, in the field of algebraic topology, the Eilenberg Moore spectral sequence addresses the calculation of the homology groups of a pullback over a fibration. The spectral sequence formulates the calculation from knowledge of the… … Wikipedia