---
tag: 6ZYQ5E545H
kind: statement
role: theorem
name: Structure theorem of divisible groups
sources:
  - wikipedia: en:472147@r1350454107
    locator: Divisible group, section Structure theorem of divisible groups
    note: adopted
    mention: wikipedia://en/472147#GII5
    sections: [Statement]
inventory: [ZWP967J2XT]
---

## Statement

Let *G* be a divisible group. Then the [[torsion subgroup|0DBWXQRXQJ]] 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|01826KMG2H]] *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*<sub>*p*</sub> for *p* ∈ **P** are uniquely determined by the group *G*.