Divisible group
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mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, specifically 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 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, 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 of its codomain; that is, implies (equivalently by contraposition, impl ...
abelian groups.


Definition

An abelian group (G, +) 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 the
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: indeed, every small abelian category can be embedded in Ab. Properties The zero object o ...
; for this reason, a divisible group is sometimes called an injective group. An abelian group is p-divisible for a prime 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 of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (for example, The set of all ...
s \mathbb Q form a divisible group under addition. * More generally, the underlying additive group of any
vector space In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
over \mathbb Q is divisible. * Every quotient of a divisible group is divisible. Thus, \mathbb Q/\mathbb Z is divisible. * The ''p''- primary component \mathbb Z /p\mathbb Z of \mathbb Q/ \mathbb Z, which is
isomorphic In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...
to the ''p''- quasicyclic group \mathbb Z ^\infty/math>, is divisible. * The multiplicative group of the
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 for ...
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, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G. Formally, given a group (mathematics), group under a binary operation  ...
of an abelian group then it is a direct summand of that abelian group. * Every abelian group can be embedded in a divisible group. Put another way, the category of abelian groups has enough injectives. * 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_ (A,T) is injective in the category of A- modules.


Structure theorem of divisible groups

Let ''G'' be a divisible group. Then the
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 (the torsion elements of ''A''). An abelian group ''A'' is called a torsion group ...
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 ''Q'' that shares certain desirable properties with the Z-module Q of all rational numbers. Specifically, if ''Q'' is a submodule ...
, Tor(''G'') is a direct summand of ''G''. So :G = \mathrm(G) \oplus G/\mathrm(G). 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(G) = \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 (mathematics), 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 ...
s ''p'' there exists I_p such that :(\mathrm(G))_p = \bigoplus_ \mathbb Z ^\infty= \mathbb Z ^\infty, where (\mathrm(G))_p is the ''p''-primary component of Tor(''G''). Thus, if P is the set of prime numbers, :G = \left(\bigoplus_ \mathbb Z ^\infty\right) \oplus \mathbb Q^. 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 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 between structures in abstract algebra, a branch of mathematics. It is defined differently but analogously for different kinds of structures. As an example, the direct sum of two abelian groups A and B is anothe ...
of injective modules is injective because the ring is
Noetherian In mathematics, the adjective Noetherian is used to describe objects that satisfy an ascending or descending chain condition on certain kinds of subobjects, meaning that certain ascending or descending sequences of subobjects must have finite leng ...
, 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.) # 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 ''Q'' that shares certain desirable properties with the Z-module Q of all rational numbers. Specifically, if ''Q'' is a submodule ...
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. Perhaps most familiar as a pr ...
domain, then the injective ''R'' modules coincide with the divisible ''R'' modules if and only if ''R'' is a Dedekind domain.


See also

* Injective object *
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. Specifically, if ''Q'' is a submodule ...


Notes


References

* With an appendix by David A. Buchsbaum; Reprint of the 1956 original * * * Chapter 13.3. * * * * * *{{citation , last1=Nicholson, first1=W. K. , last2=Yousif, first2=M. F. , title=Quasi-Frobenius rings , series=Cambridge Tracts in Mathematics , volume=158 , publisher=Cambridge University Press , place=Cambridge , year=2003 , pages=xviii+307 , isbn=0-521-81593-2 , mr=2003785 , doi=10.1017/CBO9780511546525 Abelian group theory Properties of groups