Why Abelian Group Is Cyclic?

The fundamental theorem of abelian groups states that every finitely generated abelian group is a finite direct product of primary cyclic and infinite cyclic groups. Because a cyclic group is abelian, each of its conjugacy classes consists of a single element.

Is an abelian group cyclic?

All cyclic groups are Abelian, but an Abelian group is not necessarily cyclic. All subgroups of an Abelian group are normal. In an Abelian group, each element is in a conjugacy class by itself, and the character table involves powers of a single element known as a group generator.

How do you prove abelian group is cyclic?

Thus, any two cyclic groups of orders n are isomorphic. Every cyclic group of order n is isomorphic to Zn. Since Zn is abelian under addition, so too then is the cyclic group.