Divisible group

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In
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 ...
, especially in the field of
group theory In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...
, a divisible group is an
abelian group In mathematics, an abelian group, also called a commutative group, is a group (mathematics), group in which the result of applying the group Operation (mathematics), operation to two group elements does not depend on the order in which they are w ...
in which every element can, in some sense, be divided by positive integers, or more accurately, every element is an ''n''th multiple for each positive integer ''n''. Divisible groups are important in understanding the structure of abelian groups, especially because they are the
injective In 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 ...
abelian groups.

# Definition

An abelian group $\left(G, +\right)$ is divisible if, 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 existence of $y$ for every $n$ and $g$ implies that $n G\supseteq G$, and the other direction $n G\subseteq G$ is true for every group. A third equivalent condition is that an abelian group $G$ is divisible if and only if $G$ is an
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 cohomology, in homotopy theory and in the theory of model category, mo ...
in the
category of abelian groups In mathematics, the category theory, category Ab has the abelian groups as object (category theory), objects and group homomorphisms as morphisms. This is the prototype of an abelian category: indeed, every Small category, small abelian category can ...
; for this reason, a divisible group is sometimes called an injective group. An abelian group is $p$-divisible for a
prime A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only wa ...
$p$ if for every $g \in G$, there exists $y \in G$ such that $py=g$. Equivalently, an abelian group is $p$-divisible if and only if $pG=G$.

# Examples

* The
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction (mathematics), fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ) ...
s $\mathbb Q$ form a divisible group under addition. * More generally, the underlying additive group of any
vector space In 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 ...
over $\mathbb Q$ is divisible. * Every
quotient In arithmetic, a quotient (from lat, quotiens 'how many times', pronounced ) is a quantity produced by the division (mathematics), division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to a ...
of a divisible group is divisible. Thus, $\mathbb Q/\mathbb Z$ is divisible. * The ''p''- primary component of $\mathbb Q/ \mathbb Z$, which is
isomorphic In 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 ...
to the ''p''- quasicyclic group
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a ...
s $\mathbb C^*$ is divisible. * Every existentially closed abelian group (in the model theoretic sense) is divisible.

# Properties

* If a divisible group is a
subgroup In group theory, a branch of mathematics, given a group (mathematics), 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, ...
of an abelian group then it is a
direct summand The direct sum is an operation between structures in abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include group (mathematics), gr ...
of that abelian group. * Every abelian group can be embedded in a divisible group. * Non-trivial divisible groups are not finitely generated. * Further, every abelian group can be embedded in a divisible group as an essential subgroup in a unique way. * An abelian group is divisible if and only if it is ''p''-divisible for every prime ''p''. * Let $A$ be a ring. If $T$ is a divisible group, then $\mathrm_ \left(A,T\right)$ is injective in the
category Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally *Category of being *Categories (Aristotle), ''Categories'' (Aristotle) *Category (Kant) ...
of $A$- modules.

# Structure theorem of divisible groups

Let ''G'' be a divisible group. Then the torsion subgroup Tor(''G'') of ''G'' is divisible. Since a divisible group is an
injective module In mathematics, especially in the area of abstract algebra known as module theory, an injective module is a module (mathematics), module ''Q'' that shares certain desirable properties with the Z-module Q of all rational numbers. Specifically, if '' ...
, Tor(''G'') is a
direct summand The direct sum is an operation between structures in abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include group (mathematics), gr ...
of ''G''. So :$G = \mathrm\left(G\right) \oplus G/\mathrm\left(G\right).$ 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 :$G/\mathrm\left(G\right) = \bigoplus_ \mathbb Q = \mathbb Q^.$ The structure of the torsion subgroup is harder to determine, but one can show that for all
prime number A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only wa ...
s ''p'' there exists $I_p$ such that : where $\left(\mathrm\left(G\right)\right)_p$ is the ''p''-primary component of Tor(''G''). Thus, if P is the set of prime numbers, : The cardinalities of the sets ''I'' and ''I''''p'' for ''p'' ∈ P are uniquely determined by the group ''G''.

# Injective envelope

As stated above, any abelian group ''A'' can be uniquely embedded in a divisible group ''D'' as an essential subgroup. This divisible group ''D'' is the injective envelope of ''A'', and this concept is the
injective hull In mathematics, particularly in abstract algebra, algebra, the injective hull (or injective envelope) of a module (mathematics), module is both the smallest injective module containing it and the largest essential extension of it. Injective hulls ...
in the category of abelian groups.

# Reduced abelian groups

An abelian group is said to be reduced if its only divisible subgroup is . 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.Griffith, p.7 This is a special feature of hereditary rings like the integers Z: the
direct sum The direct sum is an Operation (mathematics), operation between Mathematical structure, structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct ...
of injective modules is injective because the ring is Noetherian, 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

Several distinct definitions generalize divisible groups to divisible modules. The following definitions have been used in the literature to define a divisible module ''M'' over a ring ''R'': # ''rM'' = ''M'' for all nonzero ''r'' in ''R''. (It is sometimes required that ''r'' is not a zero-divisor, and some authors require that ''R'' is a
domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined **Domain of definition of a partial function **Natural domain of a partial function **Domain of holomorphy of a function *Dom ...
.) # For every principal left ideal ''Ra'', any
homomorphism In algebra, a homomorphism is a morphism, structure-preserving map (mathematics), map between two algebraic structures of the same type (such as two group (mathematics), groups, two ring (mathematics), rings, or two vector spaces). The word ''homo ...
from ''Ra'' into ''M'' extends to a homomorphism from ''R'' into ''M''. (This type of divisible module is also called ''principally injective module''.) # For every finitely generated left ideal ''L'' of ''R'', any homomorphism from ''L'' into ''M'' extends to a homomorphism from ''R'' into ''M''. The last two conditions are "restricted versions" of the Baer's criterion for
injective module In mathematics, especially in the area of abstract algebra known as module theory, an injective module is a module (mathematics), module ''Q'' that shares certain desirable properties with the Z-module Q of all rational numbers. Specifically, if '' ...
s. Since injective left modules extend homomorphisms from ''all'' left ideals to ''R'', injective modules are clearly divisible in sense 2 and 3. If ''R'' is additionally a domain then all three definitions coincide. If ''R'' is a principal left ideal domain, then divisible modules coincide with injective modules. Thus in the case of the ring of integers Z, which is a principal ideal domain, a Z-module (which is exactly an abelian group) is divisible if and only if it is injective. If ''R'' is a
commutative In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
domain, then the injective ''R'' modules coincide with the divisible ''R'' modules if and only if ''R'' is a
Dedekind domain In abstract algebra, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an integral domain in which every Ideal (ring_theory)#Examples and properties, nonzero proper ideal factors into a product of prime ideals. It can be shown t ...
.