In this article, we will show that every cyclic group is abelian ( or is commutative ).

**Basic Terminologies**

**Abelian Group**: A group that is commutative is know as abelian group.**Cyclic Group:**A group that contains an element**g**such that every other element of the group can be obtained by repeatedly applying the group operation on**g**. The element**g**is know as generator of the group.

**Proof**

Suppose a cyclic group **G**. Let **g** be the generator of group **G**.

Consider 2 elements **a**, **b** ∈ **G**. Since **g** is the generator, so we can write **a = g ^{n}** and

**b =**where

**g**^{m}**n, m ∈ N.**

a*b = g^{n}*g^{m}** = **g

^{n+m}= g

^{m+n}= g

^{m}*g

^{n}= b*a

Therefore, a*b = b*a. In this way, we can show that every pair of elements in group **G** are commutative. Thus, group **G** is an abelian group.

**Note:** Every cyclic group is abelian but every abelian group is not cyclic.