In
mathematicsMathematics 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 theoryIn 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 groupIn 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
injectiveIn 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 ifIn 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 objectIn 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 groupsIn 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
primeA 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
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
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
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
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-AbelianIn 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
Structure theorem of divisible groups
Let
G be a divisible group. One can easily see that the
torsion subgroupIn 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 moduleIn 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 numberA 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 subgroupIn 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 hullIn 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 ringIn 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 sumIn 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
NoetherianIn 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
moduleIn 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
ringIn 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 domainIn 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
injectiveIn 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...
.