Does abelian mean cyclic?

I'm curious to know, does the term 'abelian' inherently mean that a group is also 'cyclic'? I understand that abelian groups have the commutative property, where the order of multiplication doesn't matter. But does this automatically imply that every element in the group can be generated by a single element, which is the defining characteristic of a cyclic group? Or are there abelian groups that are not cyclic? I'd appreciate a clear explanation to help me better grasp this concept in group theory.