Quasicyclic group

Divisible group

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 are 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.

An abelian group is p-divisible for a prime p if every positive integer n and every g in G, there exists y in G such that pny = g. Equivalently, an abelian group is p-divisible if and only if pG = G.

Examples

Properties

  • If a divisible group is a subgroup of an abelian group then it is a direct summand.
  • Every abelian group can be embedded in a divisible group.
  • Further, every abelian group can be embedded in a divisible group as an essential subgroup in a unique way.
  • An abelian group is divisible if and only if it is p-divisible for every prime p.

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 I_p such that

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

where (mathrm{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)}.

Injective envelope

As stated above, any abelian group A can be uniquely embedded in a divisible group D as an essential subgroup. This divisible group D is the injective envelope of A, and this concept is the injective hull in the category of abelian groups.

Reduced abelian groups

An abelian group is said to be reduced if its only divisible subgroup is {0}. Every abelian group is the direct sum of a divisible subgroup and a reduced subgroup. In fact, there is 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 is injective.

Notes

References

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