Let G,H,K be divisible abelian groups.

(a) If G ⊕ G ≅ H⊕H, then G ≅ H [see Exercise 11].

(b) If G⊕H ≅ G⊕K, then H ≅ K [see Exercises 11.].

Data from exercise 11

Every divisible abelian group is a direct sum of copies of the rationals Q and copies of Z(p∞) for various primes p.