Divisible group

In group theory, a divisible group is an abelian group G such that for any positive n and any g in G, there exists y in G such that ny = x. One can show that G is divisible if and only if G is an injective object in the category of Z-moduless.

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 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 a injective, Tor(G) is a direct summand of G. So

.

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

.

The structure of the torsion subgroup is harder to determine, but one can show that for all prime numbers there exists such that

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

Thus, if P is the set of prime numbers,

.