In group theory, a divisible group is an abelian group G such that for any positive integer n and any g in G, there exists y in G such that ny = g. One can show that G is divisible if and only if G is an injective object in the category of abelian groups. Hence, it is also sometimes termed an injective group.

Examples

• The rational numbers Q form a divisible group under addition.
• More generally, the underlying additive group of any vector space over Q is divisible.
• Every quotient of a divisible group is divisible. Thus, Q/Z is divisible.
• The p-primary component of Q/Z which is isomorphic to the p-quasicyclic group $\backslash mathbb\; Z*$ is divisible.
• Every existentially closed group (in the model theoretic sense) is divisible.

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

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

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

$G/\backslash mathrm\{Tor\}(G)\; =\; \backslash oplus\_\{i\; \backslash in\; I\}\; \backslash mathbb\; Q\; =\; \backslash mathbb\; Q^\{(I)\}.$

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

$(\backslash mathrm\{Tor\}(G))\_p\; =\; \backslash oplus\_\{i\; \backslash in\; I\_p\}\; \backslash mathbb\; Z=\; \backslash mathbb\; Z[p^\backslash infty^\{(I\_p)\},$

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

Thus, if P is the set of prime numbers,

$G\; =\; (\backslash oplus\_\{p\; \backslash in\; \backslash mathbf\; P\}\; \backslash mathbb\; Z*^\{(I\_p)\})\; \backslash oplus\; \backslash mathbb\; Q^\{(I)\}.$

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.

Abelian group theory