In group theory, a divisible group is an abelian group G such that for any positive integer n and any g in G, there exists y in G such that ny = g. One can show that G is divisible if and only if G is an injective object in the category of abelian groups. Hence, it is also sometimes termed an injective group. Group theory is that branch of mathematics concerned with the study of groups. ... In mathematics, an abelian group, also called a commutative group, is a group such that for all a and b in G. In other words, the order of elements in a product doesnt matter. ... In mathematics, an injective module is a module Q that shares certain desirable properties with the Z-module Q of all rational numbers. ... In mathematics, the category Ab has the abelian groups as objects and group homomorphisms as morphisms. ...

Examples

• The rational numbers Q form a divisible group under addition.
• More generally, the underlying additive group of any vector space over Q is divisible.
• Every quotient of a divisible group is divisible. Thus, Q/Z is divisible.
• The p-primary component of Q/Z which is isomorphic to the p-quasicyclic group is divisible.
• Every existentially closed group (in the model theoretic sense) is divisible.

In mathematics, a rational number (commonly called a fraction) is a ratio or quotient of two integers, usually written as the vulgar fraction a/b, where b is not zero. ... In mathematics, a vector space (or linear space) is a collection of objects (known as vectors) which may be scaled and added; all linear combinations of vectors are themselves vectors. ... In mathematics, given a group G and a normal subgroup N of G, the quotient group, or factor group, of G over N is intuitively a group that collapses the normal subgroup N to the identity element. ... In mathematics, given a prime number p, a p-group is a periodic group in which each element has a power of p as its order. ... In mathematics, model theory is the study of the representation of mathematical concepts in terms of set theory, or the study of the models which underlie mathematical systems. ...

Structure theorem of divisible groups

Let G be a divisible group. One can easily see that the torsion subgroup Tor(G) of G is divisible. Since a divisible group is an injective module, Tor(G) is a direct summand of G. So In group theory, the torsion subgroup of an abelian group A is the subgroup of A consisting of all elements that have finite order. ... In mathematics, an injective module is a module Q that shares certain desirable properties with the Z-module Q of all rational numbers. ... In abstract algebra, the direct sum is a construction which combines several vector spaces (or groups, or abelian groups, or modules) into a new, bigger one. ...

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 In group theory, the torsion subgroup of an abelian group A is the subgroup of A consisting of all elements that have finite order. ...

The structure of the torsion subgroup is harder to determine, but one can show that for all prime numbers p there exists I_p such that In mathematics, a prime number (or a prime) is a natural number that has exactly two (distinct) natural number divisors, which are 1 and the prime number itself. ...

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

Thus, if P is the set of prime numbers,

Generalization

A left module M over a ring 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 is divisible if and only if it is injective. In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, where instead of requiring the scalars to lie in a field, the scalars may lie in an arbitrary ring. ... In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ... In abstract algebra, a principal ideal domain (PID) is an integral domain in which every ideal is principal (that is, generated by a single element). ... In mathematics, an injective module is a module Q that shares certain desirable properties with the Z-module Q of all rational numbers. ...