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 $(G, +)$ is divisible 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 existence of $y$ for every $n$ and $g$ implies that $n G\supset G$, and in the other direction $n G\subset G$ is true for every group. A third equivalent condition is that an abelian group $G$ is divisible if and only if $G$ 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 $p$-divisible for a prime $p$ if for every positive integer $n$ and every $g \in G$, there exists $y \in G$ such that $p^ny=g$. Equivalently, an abelian group is $p$-divisible if and only if $p^nG=G$ for every $n$.

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.
• The multiplicative group of the complex numbers $\mathbb C^*$ is divisible.
• Every existentially closed abelian group (in the model theoretic sense) is divisible.

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. Then 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^[6] 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,

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

The cardinalities of the sets I and I_p 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 {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.^[7] 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

Several distinct definitions which 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.^[8] (It is sometimes required that r is not a zero-divisor, and some authors^[9][10] require that R is a domain.)
2. For every principal left ideal Ra, any homomorphism from Ra into M extends to a homomorphism from R into M.^[11][12] (This type of divisible module is also called principally injective module.)
3. For every finitely generated left ideal L of R, any homomorphism from L into M extends to a homomorphism from R into M.^[13]

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.^[14] 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.^[14]

Notes

1. Griffith, p.6
2. Hall, p.197
3. Griffith, p.17
4. Griffith, p.19
5. Lang, p. 106
6. Kaplansky 1965.
7. Griffith, p.7
8. Feigelstock 2006.
9. Cartan & Eilenberg 1999.
10. Rotman 2009.
11. Lam 1999.
12. Nicholson & Yousif 2003.
13. Damiano 1979.
14. Lam 1999, p.70—73.

References

• Cartan, Henri; Eilenberg, Samuel (1999), Homological algebra, Princeton Landmarks in Mathematics, Princeton, NJ: Princeton University Press, pp. xvi+390, ISBN 0-691-04991-2, MR 1731415 With an appendix by David A. Buchsbaum; Reprint of the 1956 original
• Feigelstock, Shalom (2006), "Divisible is injective", Soochow J. Math. 32 (2): 241–243, ISSN 0250-3255, MR 2238765
• Griffith, Phillip A. (1970). Infinite Abelian group theory. Chicago Lectures in Mathematics. University of Chicago Press. ISBN 0-226-30870-7.
• Hall, Marshall, jr (1959). The theory of groups. New York: Macmillan. Chapter 13.3.
• Kaplansky, Irving (1965). Infinite Abelian Groups. University of Michigan Press.
• Lam, Tsit-Yuen (1999), Lectures on modules and rings, Graduate Texts in Mathematics No. 189, Berlin, New York: Springer-Verlag, ISBN 978-0-387-98428-5, MR 1653294
• Serge Lang (1984). Algebra, Second Edition. Menlo Park, California: Addison-Wesley.
• Matlis, Eben (1958). "Injective modules over Noetherian rings". Pacific Journal of Mathematics 8: 511–528. doi:10.2140/pjm.1958.8.511. ISSN 0030-8730. MR 0099360.
• Nicholson, W. K.; Yousif, M. F. (2003), Quasi-Frobenius rings, Cambridge Tracts in Mathematics 158, Cambridge: Cambridge University Press, pp. xviii+307, doi:10.1017/CBO9780511546525, ISBN 0-521-81593-2, MR 2003785