An abelian group $G$ is called divisible if for every $x \in G$ and $n \in \mathbb{Z}^+$, there exists a $y \in G$ for which $x = ny$.
The group of rational numbers $(\mathbb{Q}, +)$ is divisible, as are the quasicyclic groups $\mathbb{Z}_{p^\infty}$ for prime numbers $p$. In fact, every divisible group can be decomposed as a direct sum of copies of $\mathbb{Q}$ and quasicyclic groups for various primes.
Divisible groups are precisely the injective $\mathbb{Z}$ modules, and so are important in studying the structure of abelian groups.