A nontrivial abelian group is called divisible if for each and each nonzero , there exists an element such that .
- Prove that is divisible.
- Prove that no finite abelian group is divisible.
- Let be a rational number and . Then , so that is a th multiple. Hence is divisible.
- Let be a finite divisible abelian group. In particular, for every positive natural number , there is an element such that . Note that we may assume is minimal with respect to this property- that is, that has order . Thus contains an element of every positive order, and these must be distinct- a contradiction since is finite.