Editing Divisible group (section)

==Structure theorem of divisible groups==

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

:<math>G = \mathrm{Tor}(G) \oplus  G/\mathrm{Tor}(G).</math>

As a quotient of a divisible group, ''G''/Tor(''G'') is divisible. Moreover, it is [[torsion (algebra)|torsion-free]]. Thus, it is a vector space over '''Q''' and so there exists a set ''I'' such that

:<math>G/\mathrm{Tor}(G) = \bigoplus_{i \in I} \mathbb Q = \mathbb Q^{(I)}.</math>

The structure of the torsion subgroup is harder to determine, but one can show{{sfn|Kaplansky|1965}}{{sfn|Fuchs|1970}} that for all [[prime number]]s ''p'' there exists <math>I_p</math> such that

:<math>(\mathrm{Tor}(G))_p = \bigoplus_{i \in I_p} \mathbb Z[p^\infty] = \mathbb Z[p^\infty]^{(I_p)},</math>

where <math>(\mathrm{Tor}(G))_p</math> is the ''p''-primary component of Tor(''G'').

Thus, if '''P''' is the set of prime numbers,

:<math>G = \left(\bigoplus_{p \in \mathbf P} \mathbb Z[p^\infty]^{(I_p)}\right) \oplus \mathbb Q^{(I)}.</math>

The cardinalities of the sets ''I'' and ''I''<sub>''p''</sub> for ''p''&nbsp;&isin;&nbsp;'''P''' are uniquely determined by the group ''G''.