Question

every abelian group is cyclic or every cyclic group is abelian?

Answer 1

every cyclic group is abelian

Answer 2

Note :

• Group : An algebraic system (G,*) is said to be a group if the following condition are satisfied :

1. G is closed under *
2. G is associative under *
3. G has a unique identity element
4. Every element of G has a unique inverse in G

• Abelian group : If a group (G,*) also holds commutative property , then it is called commutative group or abelian group , ie . if x*y = y*x ∀ x , y ∈ (G,*) , then the group G is said to be abelian .

Solution :

To prove :

Every cyclic group is an abelian group .

Proof :

Let G = < a > be a cyclic group with generator a .

Let x and y be any two elements of G . Then there exist integers r and s such that x = aʳ and y = aˢ.

Now ,

→ xy = aʳaˢ

→ xy = aʳ⁺ˢ

→ xy = aˢ⁺ʳ

→ xy = aˢaʳ

→ xy = yx

Thus , we have xy = yx ∀ x , y ∈ G .

→ G is an abelian group .

Hence ,

Every cyclic group is an abelian group .

But the converse is not true , ie . not every abelian group is cyclic . An abelian group may or may not be cyclic .

Examples :

• The Klein's 4 group , K₄ is abelian but not cyclic .
• (R,+) and (Q,+) are abelian but not cyclic .