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 abelian groups. Hence, it is also sometimes termed an injective group.

[edit] Examples

  • The rational numbers Q form a divisible group under addition.
  • More generally, the underlying additive group of any vector space over Q is divisible.
  • 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 \mathbb Z[p^\infty] is divisible.
  • Every existentially closed group (in the model theoretic sense) is divisible.

[edit] 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

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) = \oplus_{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 Ip such that

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

where (Tor(G))p is the p-primary component of Tor(G).

Thus, if P is the set of prime numbers,

G = (\oplus_{p \in \mathbf P} \mathbb Z[p^\infty]^{(I_p)}) \oplus \mathbb Q^{(I)}.

[edit] Generalization

A left module M over a ring R is called a divisible module if rM=M for all nonzero r in R. Thus a divisible abelian group is simply a divisible Z-module. A module over a principal ideal domain is divisible if and only if it is injective.

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