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...

, a slender group is a torsion-free abelian group

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...

 that is "small" in a sense that is made precise in the definition below.

Definition

Let Z^N denote the Baer–Specker group

Baer–Specker group

In mathematics, in the field of group theory, the Baer–Specker group, or Specker group, is an example of an infinite Abelian group which is a building block in the structure theory of such groups.-Definition:...

, that is, the group of all integer sequence

Integer sequence

In mathematics, an integer sequence is a sequence of integers.An integer sequence may be specified explicitly by giving a formula for its nth term, or implicitly by giving a relationship between its terms...

s, with termwise addition. For each n in N, let e_n be the sequence with n-th term equal to 1 and all other terms 0.

A torsion-free abelian group G is said to be slender if every homomorphism

Group homomorphism

In mathematics, given two groups and , a group homomorphism from to is a function h : G → H such that for all u and v in G it holds that h = h \cdot h...

 from Z^N into G maps all but finitely many of the e_n to the identity element.

Examples

Every free abelian group

Free abelian group

In abstract algebra, a free abelian group is an abelian group that has a "basis" in the sense that every element of the group can be written in one and only one way as a finite linear combination of elements of the basis, with integer coefficients. Hence, free abelian groups over a basis B are...

 is slender.

Q is not slender: any mapping of the e_n into Q extends to a homomorphism from the free subgroup generated by the e_n, and as Q is injective

Injective module

In mathematics, especially in the area of abstract algebra known as module theory, an injective module is a module Q that shares certain desirable properties with the Z-module Q of all rational numbers...

 this homomorphism extends over the whole of Z^N. Therefore, a slender group must be reduced.

Every countable

Countable set

In mathematics, a countable set is a set with the same cardinality as some subset of the set of natural numbers. A set that is not countable is called uncountable. The term was originated by Georg Cantor...

 reduced torsion-free abelian group is slender, so every proper subgroup of Q is slender.

Properties

• A torsion-free abelian group is slender 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....
  
   it is reduced and contains no copy of the Baer–Specker group and no copy of the p-adic integers for any p.
• Direct sums of slender groups are also slender.
• Subgroups of slender groups are slender.
• Every homomorphism from Z^N into a slender group factors through Zⁿ for some natural number n.