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

Definition: A divisible group G is an abelian group such that for any x \in G and n \geq 1 there exists y \in G satisfying x=ny.

Definition: An injective group D is an abelian group such that for any abelian groups A and B and any morphisms j : A \hookrightarrow B and \varphi : A \to D there exists a morphism \eta : B \to D so that \eta \circ j=\varphi.

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

Proof. Let D be a divisible group. For x \in D and n \geq 1, let \eta be the morphism given by the definition where A= \langle x \rangle and B is a divisible group containing A (eg. A= \mathbb{Q} if x has infinite order or A is a direct sum of Prüfer groups otherwise).

Because B is divisible, there exists y \in B such that j(x)=ny, hence x= \eta \circ j(x)=n \cdot \eta(y). Therefore, D is divisible.

Let D be a divisible group. Let A,B be two abelian groups and j : A \hookrightarrow B, \varphi : A \to D be two morphisms. Without loss of generality, we suppose that A is a subgroup of B and that j is just the canonical injection.

Let \mathcal{P} be the set of pairs (C,\theta) where A \leq C \leq B and \theta : C \to D extends \varphi, ordered by:

(C_1,\theta_1) \prec (C_2,\theta_2) if C_1 \leq C_2 and \theta_2 extends \theta_1.

Notice that (\mathcal{P},\prec) is inductive since every chain \{(C_i,\theta_i) \mid i \in I\} is bounded above by \left( \bigcup\limits_{i \in I} C_i , \bigcup\limits_{i \in I} \theta_i \right) \in \mathcal{P}. According to Zorn’s lemma, there exists a maximal element (C,\theta) of \mathcal{P}.

By contradiction, suppose that C \subsetneq B and let b \in B \backslash C.

  • Case 1: C \cap \langle b \rangle= \langle nb \rangle where n \geq 1.

Because D is divisible there exists d \in D such that \theta(nb)=nd. Introduce:

\psi : \left\{ \begin{array}{ccc} \langle C,b \rangle & \to & D \\ c+mb & \mapsto & \theta(c)+md \end{array} \right. where c \in C and 0 \leq m < n.

First, \psi is well-defined since any element of \langle C,b \rangle can be uniquely written as c+mb where c \in C and 0 \leq m <n. Indeed, suppose that c+mb=c'+m'b where c,c' \in C and 0 \leq m,m' < n. Then (m-m')b=c'-c \in C \cap \langle b \rangle =\langle nb \rangle hence m=m' and c=c' since -n<m-m'<n.

Secondly, \psi is a morphism: if c,c' \in C and 0 \leq m,m' <n, writing the division m-m'=kn+p, we have:

\begin{array}{ll} \psi((c+mb)-(c'+m'b)) & = \psi((c-c'+knb)+pb) = \theta(c-c'+knb)+pd \\ \\ & = \theta(c)-\theta(c') +k \underset{nd}{\underbrace{\theta(nb)}}+pd \\ \\ & = \theta(c)-\theta(c')+ \underset{m-m'}{\underbrace{(kn+p)}}d \\ \\ & = ( \theta(c)+md)-(\theta(c')+m'd) \\ \\ & = \psi(c+mb)-\psi(c'+m'b) \end{array}

  • Case 2: C \cap \langle b \rangle = \{0\}.

Taking any d \in B, we define in a similar way:

\psi : \left\{ \begin{array}{ccc} \langle C,b \rangle & \to & D \\ c+mb & \mapsto & \theta(c)+md \end{array} \right. where c \in C and m \in \mathbb{Z}.

Again, \psi is a well-defined homomorphism.

Finally, (\langle C,b \rangle,\psi) \in \mathcal{P} contradicts the maximality of (C,\theta). Consequently, C=B and it is sufficient to set \eta:= \theta to conclude that D is injective. \square

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

\displaystyle \sum\limits_{j \in J} n_{ij}x_j=a_i, i \in I

where a_i \in A, n_{ij} \in \mathbb{Z} and where \{ n_{ij} \neq 0 \mid j \in J\} is finite for each i \in I. Formally, f_i := \sum\limits_{j \in J} n_{ij}x_j can be viewed as an element of the abelian free group F over \{x_j \mid j \in J\}.

We say that the system of equations is compatible over A if a linear combination of f_i‘s leads to a linear combination of a_i‘s, ie. if f_k= \sum\limits_{m \in M} f_m then a_k= \sum\limits_{m \in M} a_m (where M is finite). Clearly, if the system has a solution over A then it is necessarily compatible.

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

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

nx_{a,n} =a with n \geq 1 and a \in A

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

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

\displaystyle \sum\limits_{j \in J} n_{ij}x_j=a_i with i \in I and n_{ij} \in \mathbb{Z}.

As above, let X be the free abelian group over \{ x_j \mid j \in J\} and Y be the subgroup generated by \{ f_i:= \sum\limits_{j \in J} n_{ij}x_j \mid i \in I\}.

Because the system is compatible, f_i \mapsto a_i extends to a homomorphism \chi : Y \to A. According to theorem 1, A is an injective group, so \chi extends to a homomorphism \eta : X \to A. Finally, \{ \eta(x_i) \mid i \in I\} is a solution to the system. \square

Corollary: A system of equations over a divisible group A 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 B of an abelian group A is a direct summand if, and only if, every system of equations over B solvable in A is solvable in B.

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

Conversely, suppose that every system of equations over B solvable in A is solvable in B. For any coset u \in A/B let a(u) \in A be a representant and consider the following compatible system of equations over B:

x_u+x_v-x_{u+v} = a(u)+a(v)-a(u+v) \in B with u,v \in A/B.

By assumption, there exists a solution \{b(u) \mid u \in A/B\}. Let C= \{ a(u)-b(u) \mid u \in A/B\}; by construction, C is a subgroup of A. Moreover, A=B+C and B \cap C = \{0\}. Therefore, B is a direct summand. \square

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

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