Structure theorem of divisible groups

Statement

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

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

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

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

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.