Divisible group
From Exampleproblems
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
![Z[p^\infty]](/wiki/images/math/c/c/0/cc01a12bea9d9843a69d61965844f09c.png)
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
.
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
.
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
![(Tor(G))_p = \oplus_{i \in I_p} Z[p^\infty] = Z[p^\infty]^{(I_p)},](/wiki/images/math/2/a/d/2ad9fdcdd84066c2d327316107694f0d.png)
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 P} Z[p^\infty]^{(I_p)}) \oplus Q^{(I)}](/wiki/images/math/7/6/b/76b1b2b9109ff84310249358b853ad6a.png)
.
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.
