Divisible group
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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 <math>Z[p^\infty]<math> 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
- <math>G = Tor(G) \oplus G/Tor(G)<math>.
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
- <math>G = \oplus_{i \in I} Q = Q^{(I)}<math>.
The structure of the torsion subgroup is harder to determine, but one can show that for all prime numbers p there exists <math>I_p<math> such that
- <math>(Tor(G))_p = \oplus_{i \in I_p} Z[p^\infty] = Z[p^\infty]^{(I_p)},<math>
where <math>(Tor(G))_p<math> is the p-primary component of Tor(G).
Thus, if P is the set of prime numbers,
- <math>G = (\oplus_{p \in P} Z[p^\infty]^{(I_p)}) \oplus Q^{(I)}<math>.