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  "body": "\n## Statement\n\nLet *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\n\n$$G = \\mathrm{Tor}(G) \\oplus G/\\mathrm{Tor}(G).$$\n\nAs 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\n\n$$G/\\mathrm{Tor}(G) = \\bigoplus_{i \\in I} \\mathbb Q = \\mathbb Q^{(I)}.$$\n\nThe 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\n\n$$(\\mathrm{Tor}(G))_p = \\bigoplus_{i \\in I_p} \\mathbb Z[p^\\infty] = \\mathbb Z[p^\\infty]^{(I_p)},$$\n\nwhere $(\\mathrm{Tor}(G))_p$ is the *p*-primary component of Tor(*G*).\n\nThus, if **P** is the set of prime numbers,\n\n$$G = \\left(\\bigoplus_{p \\in \\mathbf P} \\mathbb Z[p^\\infty]^{(I_p)}\\right) \\oplus \\mathbb Q^{(I)}.$$\n\nThe cardinalities of the sets *I* and *I*\u003csub\u003e*p*\u003c/sub\u003e for *p* ∈ **P** are uniquely determined by the group *G*.\n",
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