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Slender group
 
 
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In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, a slender group is a torsion-free abelian group
Abelian group

An abelian group, also called a commutative group, is a group satisfying the requirement that the product of elements does not depend on their order ....
 that is "small" in a sense that is made precise in the definition below.

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....
 is slender.

Q is not slender: any mapping of the en into Q extends to a homomorphism from the free subgroup generated by the en, 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 ZN.






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In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, a slender group is a torsion-free abelian group
Abelian group

An abelian group, also called a commutative group, is a group satisfying the requirement that the product of elements does not depend on their order ....
 that is "small" in a sense that is made precise in the definition below.

Definition


Let ZN 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....
, 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 en 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 group and , a group homomorphism from to is a function h : G ? H such that for all u and v in G it holds that...
 from ZN into G maps all but finitely many of the en 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....
 is slender.

Q is not slender: any mapping of the en into Q extends to a homomorphism from the free subgroup generated by the en, 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 ZN. 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. The term was originated by Georg Cantor; it stems from the fact that the natural numbers are often called counting numbers....
 reduced torsion-free abelian group is slender, so every proper subgroup of Q is slender.

Properties

  • A torsion-free group is slender if and only if
    If and only if

    If and only if, in logic and fields that rely on it such as mathematics and philosophy, 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.