Following L. Fuchs’ book *Infinite Abelian Groups*, we give in this note three equivalent definitions of divisibility of a group.

**Definition:** A divisible group is an abelian group such that for any and there exists satisfying .

**Definition:** An injective group is an abelian group such that for any abelian groups and and any morphisms and there exists a morphism so that .

**Theorem 1:** A group is injective if, and only if, it is divisible.

**Proof.** Let be a divisible group. For and , let be the morphism given by the definition where and is a divisible group containing (eg. if has infinite order or is a direct sum of Prüfer groups otherwise).

Because is divisible, there exists such that , hence . Therefore, is divisible.

Let be a divisible group. Let be two abelian groups and , be two morphisms. Without loss of generality, we suppose that is a subgroup of and that is just the canonical injection.

Let be the set of pairs where and extends , ordered by:

if and extends .

Notice that is inductive since every chain is bounded above by . According to Zorn’s lemma, there exists a maximal element of .

By contradiction, suppose that and let .

- Case 1: where .

Because is divisible there exists such that . Introduce:

where and .

First, is well-defined since any element of can be uniquely written as where and . Indeed, suppose that where and . Then hence and since .

Secondly, is a morphism: if and , writing the division , we have:

- Case 2: .

Taking any , we define in a similar way:

where and .

Again, is a well-defined homomorphism.

Finally, contradicts the maximality of . Consequently, and it is sufficient to set to conclude that is injective.

In an abelian group , a system of equations can be written as

,

where , and where is finite for each . Formally, can be viewed as an element of the abelian free group over .

We say that the system of equations is compatible over if a linear combination of ‘s leads to a linear combination of ‘s, ie. if then (where is finite). Clearly, if the system has a solution over then it is necessarily compatible.

**Theorem 2:** Let be an abelian group. Every compatible system of equations over has a solution in if, and only if, is a divisible group.

**Proof.** First, notice that the following system of equations is compatible over :

with and

Therefore, if every compatible system of equations has a solution, then is a divisible group.

Conversely suppose that is a divisible group and consider a compatible system of equations overs :

with and .

As above, let be the free abelian group over and be the subgroup generated by .

Because the system is compatible, extends to a homomorphism . According to theorem 1, is an injective group, so extends to a homomorphism . Finally, is a solution to the system.

**Corollary:** A system of equations over a divisible group has a solution if, and only if, every finite subsystem has a solution.

Just for fun, let us mention the following property:

**Property:** A subgroup of an abelian group is a direct summand if, and only if, every system of equations over solvable in is solvable in .

**Proof.** If is a direct summand of , then the projection on of any solution of a system of equations over is a solution in .

Conversely, suppose that every system of equations over solvable in is solvable in . For any coset let be a representant and consider the following compatible system of equations over :

with .

By assumption, there exists a solution . Let ; by construction, is a subgroup of . Moreover, and . Therefore, is a direct summand.

**Corollary:** Any divisible subgroup of an abelian group is a direct summand.