Divisible group


In mathematics, especially 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.

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 in 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 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 if for every, there exists such that. Equivalently, an abelian group is -divisible if and only if.

Examples

Let G be a divisible group. Then the torsion subgroup Tor of G is divisible. Since a divisible group is an injective module, Tor is a direct summand of G. So
As a quotient of a divisible group, G/Tor 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 numbers p there exists such that
where is the p-primary component of Tor.
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.

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. This is a special feature of hereditary rings like the integers Z: the direct sum of injective modules is injective because the ring is Noetherian, 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:
  1. rM = M for all nonzero r in R.
  2. For every principal left ideal Ra, any homomorphism from Ra into M extends to a homomorphism from R into M.
  3. 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 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. Thus in the case of the ring of integers Z, which is a principal ideal domain, a Z-module 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.