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.

1 Definition
2 Examples
3 Properties
4 Structure theorem of divisible groups
5 Injective envelope
6 Reduced abelian groups
7 Generalization
8 Notes
9 References

Definition

An abelian group G is divisible if and only if for every positive integer n and every g in G, there exists y in G such that ny = g.^[1] 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 object in the category of abelian groups, so a divisible group is sometimes called an injective group.

An abelian group is p-divisible for a prime 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 numbers $\mathbb Q$ form a divisible group under addition.
More generally, the underlying additive group of any vector space over $\mathbb Q$ is divisible.
Every quotient of a divisible group is divisible. Thus, $\mathbb Q/\mathbb Z$ is divisible.
The p-primary component $\mathbb Z[1/p]/\mathbb Z$ of $\mathbb Q/ \mathbb Z$ , which is isomorphic to the p-quasicyclic group $\mathbb Z[p^\infty]$ is divisible.
Every existentially closed group (in the model theoretic sense) is divisible.
The space of orientation-preserving isometries of $\mathbb R^2$ 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-Abelian divisible group.

Properties

If a divisible group is a subgroup of an abelian group then it is a direct summand.^[2]
Every abelian group can be embedded in a divisible group.^[3]
Non-trivial divisible groups are not finitely generated.
Further, every abelian group can be embedded in a divisible group as an essential subgroup in a unique way.^[4]
An abelian group is divisible if and only if it is p-divisible for every prime p.
Let $A$ be a ring. If $T$ is a divisible group, then $\mathrm{Hom}_\mathbf{Z} (A,T)$ is injective in the category of $A$ -modules.^[5]

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

$G = \mathrm{Tor}(G) \oplus G/\mathrm{Tor}(G).$

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

$G/\mathrm{Tor}(G) = \oplus_{i \in I} \mathbb Q = \mathbb Q^{(I)}.$

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

$(\mathrm{Tor}(G))_p = \oplus_{i \in I_p} \mathbb Z[p^\infty] = \mathbb Z[p^\infty]^{(I_p)},$

where $(\mathrm{Tor}(G))_p$ is the p-primary component of Tor(G).

Thus, if P is the set of prime numbers,

$G = (\oplus_{p \in \mathbf P} \mathbb Z[p^\infty]^{(I_p)}) \oplus \mathbb Q^{(I)}.$

Injective envelope

Main article: 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 {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.^[6] 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 (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.

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.

Notes

^ Griffith, p.6
^ Hall, p.197
^ Griffith, p.17
^ Griffith, p.19
^ Lang, p. 106
^ Griffith, p.7

References

Phillip A. Griffith (1970). Infinite Abelian group theory. Chicago Lectures in Mathematics. University of Chicago Press. ISBN 0-226-30870-7.
Marshall Hall jr (1959). The theory of groups. New York: Macmillan. Chapter 13.3.
Serge Lang (1984). Algebra, Second Edition. Menlo Park, California: Addison-Wesley.
Eben Matlis (1958). "Injective modules over Noetherian rings". Pacific Journal of Mathematics 8: 511–528. ISSN 0030-8730. MR 0099360. http://projecteuclid.org/getRecord?id=euclid.pjm/1103039896.