mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, a slender group is a torsion-free

abelian group In mathematics, 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 the order in which they are written. That is, the group operation is commut ...

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

Definition

Let Z^N denote the

Baer–Specker group In mathematics, in the field of group theory, the Baer–Specker group, or Specker group, named after Reinhold Baer and Ernst Specker, is an example of an infinite Abelian group which is a building block in the structure theory of such groups. Defi ...

, that is, the

group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...

of all

integer sequence In mathematics, an integer sequence is a sequence (i.e., an ordered list) of integers. An integer sequence may be specified ''explicitly'' by giving a formula for its ''n''th term, or ''implicitly'' by giving a relationship between its terms. For ...

s, with termwise addition. For each

natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''Cardinal n ...

''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 In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "same" ...

from Z^N into ''G'' maps all but finitely many of the ''e''_''n'' to the

identity element In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures su ...

Examples

Every

free abelian group In mathematics, a free abelian group is an abelian group with a basis. Being an abelian group means that it is a set with an addition operation that is associative, commutative, and invertible. A basis, also called an integral basis, is a subse ...

is slender. The additive group of

rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ration ...

s Q is not slender: any mapping of the ''e''_''n'' into Q extends to a homomorphism from the free

subgroup In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup ...

generated by the ''e''_''n'', and as Q is

injective In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositiv ...

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

countable In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers; ...

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 In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bicondi ...

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 Z^''n'' for some natural number ''n''.

References

* . * * * Properties of groups Abelian group theory {{Abstract-algebra-stub