Definition of quasi-cyclic and full rational groups

Question

In Unit Groups of Classical Rings by Karpilovsky, p.96, we can see this theorem:

Let $G$ be a divisible abelian group. Then $G$ is a direct product quasi-cyclic and full rational groups.

I want to know the definition of quasi cyclic group and the full rational group. Are they equal to $\mathbb Z_{p^\infty}$ and $\mathbb{Q}$?

Answer 1

Yes. The torsion subgroup $T$ of $G$ is a direct sum of copies of $\mathbb{Z}_{p^\infty}$, for various primes $p$, and $G/T$ is a rational vector space, so it is isomorphic to a direct sum of copies of $\mathbb{Q}$. Since $T$ is a direct summand of $G$, therefore, $G$ is isomorphic to $T\oplus G/T$.