It is well known fact that every cyclic group is abelian. But the converse is not true always. i.e. There exists groups that are abelian but not cyclic.
Uncountable abelian groups
The groups such as and are abelian groups. Since any infinite cyclic group is isomorphic to and hence is countable, the above mentioned groups cannot be cylic as they are uncountable. In fact any uncountable group is non-cyclic.
Finite abelian groups
The Fundamental theorem of finite abelian groups states that ” Every finite abelian group is isomorphic to direct product of cyclic groups”. Also is cyclic if and only if . Using above facts, we can conclude that any abelian group which is isomorphic to with , is non-cyclic but abelian.
Countable abelian groups
- is abelian but not cyclic. Note that here every element is of infinite order.
- is abelian but not cyclic. For, if suppose it was cyclic, then will be isomorphic to . But contains a non-trivial finite order element whereas does not.
- is abelian since is abelian. Every element of this group is of finite order since where . Also this group is infinite since is an infinite subset of . Hence is an infinite non-cyclic abelian group.