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Divisible group
From Wikipedia, the free encyclopedia
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. An equivalent condition is: for any positive integer n, nG=G (since the first condition implies one set containment and the other is always true.) 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.
- 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]](http://upload.wikimedia.org/math/6/5/6/656d5267d5a0ad453d6601dbcd02b0d8.png)
is divisible. - Every existentially closed group (in the model theoretic sense) is divisible.
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
![(\mathrm{Tor}(G))_p = \oplus_{i \in I_p} \mathbb Z[p^\infty] = \mathbb Z[p^\infty]^{(I_p)},](http://upload.wikimedia.org/math/1/b/3/1b31bd9bcb69ec23c8dd306aa180d522.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 \mathbf P} \mathbb Z[p^\infty]^{(I_p)}) \oplus \mathbb Q^{(I)}.](http://upload.wikimedia.org/math/b/a/b/bab64aa4a674220cb8e20dc6d5e2caac.png)
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.