Divisible group

# Divisible group

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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...

, especially in the field of 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...

, a divisible group is an 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...

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. Divisible groups are important in understanding the structure of abelian groups, especially because they are the 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...

abelian groups.

## Definition

An abelian group G is divisible 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....

for every positive integer n and every g in G, there exists y in G such that ny = g. An equivalent condition is: for any positive integer n, nG = G, since the first condition implies one set containment and the other is always true. An abelian group G is divisible if and only if G is an injective object
Injective object
In mathematics, especially in the field of category theory, the concept of injective object is a generalization of the concept of injective module. This concept is important in homotopy theory and in theory of model categories...

in the category of abelian groups
Category of abelian groups
In mathematics, the category Ab has the abelian groups as objects and group homomorphisms as morphisms. This is the prototype of an abelian category....

, so a divisible group is sometimes called an injective group.

An abelian group is p-divisible for a prime
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...

p if for every positive integer n and every g in G, there exists y in G such that pny = g. Equivalently, an abelian group is p-divisible if and only if pG = G.

## Examples

• The 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 form a divisible group under addition.
• More generally, the underlying additive group of any vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...

over is divisible.
• Every quotient
Quotient group
In mathematics, specifically group theory, a quotient group is a group obtained by identifying together elements of a larger group using an equivalence relation...

of a divisible group is divisible. Thus, is divisible.
• The p-primary component  of , which is isomorphic to the p-quasicyclic group  is divisible.
• Every existentially closed group (in the model theoretic
Model theory
In mathematics, model theory is the study of mathematical structures using tools from mathematical logic....

sense) is divisible.
• The space of orientation-preserving isometries of is divisible. This is because each such isometry is either a translation or a rotation about a point, and in either case the ability to "divide by n" is plainly present. This is the simplest example of a non-Abelian
Abelian
In mathematics, Abelian refers to any of number of different mathematical concepts named after Niels Henrik Abel:- Group theory :*Abelian group, a group in which the binary operation is commutative...

divisible group.

## Properties

• If a divisible group is a subgroup of an abelian group then it is a direct summand.
• Every abelian group can be embedded in a divisible group.
• Non-trivial divisible groups are not finitely generated
Finitely generated abelian group
In abstract algebra, an abelian group is called finitely generated if there exist finitely many elements x1,...,xs in G such that every x in G can be written in the formwith integers n1,...,ns...

.
• Further, every abelian group can be embedded in a divisible group as an essential subgroup
Essential subgroup
In mathematics, especially in the area of algebra studying the theory of abelian groups, an essential subgroup is a subgroup that determines much of the structure of its containing group...

in a unique way.
• An abelian group is divisible if and only if it is p-divisible for every prime p.
• Let be a ring. If is a divisible group, then is injective in the category of -modules.

## Structure theorem of divisible groups

Let G be a divisible group. One can easily see that the torsion subgroup
Torsion subgroup
In the theory of abelian groups, the torsion subgroup AT of an abelian group A is the subgroup of A consisting of all elements that have finite order...

Tor(G) of G is divisible. Since a divisible group is an injective module
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...

, Tor(G) is a direct summand of G. So

As a quotient of a divisible group, G/Tor(G) is divisible. Moreover, it is torsion-free. Thus, it is a vector space over Q and so there exists a set I such that

The structure of the torsion subgroup is harder to determine, but one can show that for all prime number
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...

s p there exists such that

where is the p-primary component of Tor(G).

Thus, if P is the set of prime numbers,

## Injective envelope

As stated above, any abelian group A can be uniquely embedded in a divisible group D as an essential subgroup
Essential subgroup
In mathematics, especially in the area of algebra studying the theory of abelian groups, an essential subgroup is a subgroup that determines much of the structure of its containing group...

. This divisible group D is the injective envelope of A, and this concept is the injective hull
Injective hull
In mathematics, especially in the area of abstract algebra known as module theory, the injective hull of a module is both the smallest injective module containing it and the largest essential extension of it...

in the category of abelian groups.

## Reduced abelian groups

An abelian group is said to be reduced if its only divisible subgroup is {0}. Every abelian group is the direct sum of a divisible subgroup and a reduced subgroup. In fact, there is a unique largest divisible subgroup of any group, and this divisible subgroup is a direct summand. This is a special feature of hereditary ring
Hereditary ring
In mathematics, especially in the area of abstract algebra known as module theory, a ring R is called hereditary if all submodules of projective modules over R are again projective...

s like the integers Z: the direct sum
Direct sum of modules
In abstract algebra, the direct sum is a construction which combines several modules into a new, larger module. The result of the direct summation of modules is the "smallest general" module which contains the given modules as submodules...

of injective modules is injective because the ring is Noetherian
Noetherian ring
In mathematics, more specifically in the area of modern algebra known as ring theory, a Noetherian ring, named after Emmy Noether, is a ring in which every non-empty set of ideals has a maximal element...

, and the quotients of injectives are injective because the ring is hereditary, so any submodule generated by injective modules is injective. The converse is a result of : if every module has a unique maximal injective submodule, then the ring is hereditary.

A complete classification of countable reduced periodic abelian groups is given by Ulm's theorem.

## Generalization

A left module
Module (mathematics)
In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...

M over a ring
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...

R is called a divisible module if rM=M for all nonzero r in R . Thus a divisible abelian group is simply a divisible Z-module. A module over a principal ideal domain
Principal ideal domain
In abstract algebra, a principal ideal domain, or PID, is an integral domain in which every ideal is principal, i.e., can be generated by a single element. More generally, a principal ideal ring is a nonzero commutative ring whose ideals are principal, although some authors refer to PIDs as...

is divisible if and only if it 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...

.