Divisible group

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 Z-modules (abelian groups).

Examples

Q is divisible, as additive abelian group
More generally, every vector space over Q has a divisible underlying group.
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 $Z[p^\infty]$ 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

$G = Tor(G) \oplus G/Tor(G)$ .

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

$G = \oplus_{i \in I} Q = Q^{(I)}$ .

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

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

where $(T o r (G)) p$ is the p-primary component of Tor(G).

Thus, if P is the set of prime numbers,

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

Divisible group

Examples

Structure theorem of divisible groups

See also: Divisible group, Abelian group, Category theory, Direct summand, Group theory, Injective module, Injective object, Model theory, Module (mathematics), Prime number