In mathematics, specifically in the field of group theory, 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 nth 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.
[1] An equivalent condition is: for any positive integer
A third equivalent condition is that an abelian group
is an injective object in the category of abelian groups; for this reason, a divisible group is sometimes called an injective group.
Equivalently, an abelian group is
Then the torsion subgroup Tor(G) of G is divisible.
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[6][7] that for all prime numbers p there exists
Thus, if P is the set of prime numbers, The cardinalities of the sets I and Ip for p ∈ P are uniquely determined by the group G. 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.
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.
The converse is a result of (Matlis 1958): 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.
The following definitions have been used in the literature to define a divisible module M over a ring R: The last two conditions are "restricted versions" of the Baer's criterion for injective modules.
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.
[13] 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 domain, then the injective R modules coincide with the divisible R modules if and only if R is a Dedekind domain.