Divisible group Totally Explained

Everything about Divisible Group totally explained

In mathematics, especially in the field of group theory, a divisible group is an abelian group in which every element can, in some sense, be divided by positive integers, or more accurately, every element is an nth multiple for each positive integer n. Divisible groups are important in understanding the structure of abelian groups, especially because they're the injective abelian groups.

Definition

An abelian group G is divisible if and only if for every positive integer n and every 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. An abelian group G is divisible if and only if G is an injective object in the category of abelian groups, so a divisible group is sometimes called an injective group.

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 Z[1/p]/Z 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.

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 ť

G = mathrm. Every abelian group is the direct sum of a divisible subgroup and a reduced subgroup.  In fact, there's a unique largest divisible subgroup of any group, and this divisible subgroup is a direct summand.

This is a special feature of hereditary rings like the integers Z: the direct sum of injective modules is injective because the ring is Noetherian, and the quotients of injectives are injective because the ring is hereditary, so any submodule generated by injective modules is injective. The converse is a result of : if every module has a unique maximal injective submodule, then the ring is hereditary.

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's injective.

Further Information

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