In
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 in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
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 abelian groups.
Definition
An abelian group
is divisible if, for every positive integer
and every
, there exists
such that
. An equivalent condition is: for any positive integer
,
, since the existence of
for every
and
implies that
, and the other direction
is true for every group. A third equivalent condition is that an abelian group
is divisible if and only if
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 categori ...
in the
category of abelian groups; for this reason, a divisible group is sometimes called an injective group.
An abelian group is
-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 ways ...
if for every
, there exists
such that
. Equivalently, an abelian group is
-divisible if and only if
.
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 (e.g. ). The set of all rat ...
s
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 whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
over
is divisible.
* Every
quotient
In arithmetic, a quotient (from lat, quotiens 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to as the integer part of a ...
of a divisible group is divisible. Thus,
is divisible.
* The ''p''-
primary component
In mathematics, the Lasker–Noether theorem states that every Noetherian ring is a Lasker ring, which means that every ideal can be decomposed as an intersection, called primary decomposition, of finitely many ''primary ideals'' (which are relate ...
of
, which is
isomorphic to the ''p''-
quasicyclic group