A finite direct product is divisible if and only if the direct factors are divisible

Let A and B be nontrivial abelian groups. Prove that A \times B is divisible if and only if A and B are divisible.


(\Rightarrow) Suppose A \times B is divisible, and let a \in A. Since A \times B is divisible, for every k \in \mathbb{Z}^+ there exists an element (x,y) \in A \times B such that (x,y)^k = (x^k,y^k) = (a,1). Thus for every k there exists x \in A such that x^k = a; hence A is divisible. By a similar argument, B is divisible.

(\Leftarrow) Suppose A and B are divisible, and let (a,b) \in A \times B. Let k \in \mathbb{Z}^+. Since A and B are divisible, there exist x \in A and y \in B such that x^k = a and y^k = b; then (x,y)^k = (x^k,y^k) = (a,b). Thus A \times B is divisible.

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