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 :
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 :
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.