Show that every cyclic group is abelian

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

Basic Terminologies

1. Abelian Group: A group that is commutative is know as abelian group.
2. 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 = gⁿ and b = g^m where n, m ∈ N.

a*b = gⁿ*g^m = g^n+m = g^m+n = g^m*gⁿ = 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.