In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, in the realm of abelian

Abelian group

In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...

 group theory

Group theory

In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...

, an abelian group

Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...

 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 number

Rational number

In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...

s.

Some properties of cotorsion groups:

• Any quotient
  Quotient
  In mathematics, a quotient is the result of division. For example, when dividing 6 by 3, the quotient is 2, while 6 is called the dividend, and 3 the divisor. The quotient further is expressed as the number of times the divisor divides into the dividend e.g. The quotient of 6 and 2 is also 3.A...
  
   of a cotorsion group is cotorsion.
• A direct product of groups
  Direct product of groups
  In the mathematical field of group theory, the direct product is an operation that takes two groups and and constructs a new group, usually denoted...
  
   is cotorsion if and only if
  If and only if
  In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
  
   each factor is.
• Every divisible group
  Divisible group
  In mathematics, especially in the field of group theory, a divisible group is an abelian group in which every element can, in some sense, be divided by positive integers, or more accurately, every element is an nth multiple for each positive integer n...
  
   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
  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:...
  
  .
• Ulm subgroups of cotorsion groups are cotorsion and Ulm factors of cotorsion groups are algebraically compact.

External links

• Springer Online Reference on cotorsion groups