Math, asked by shobhanshubaibil, 10 months ago

Prove that every cyclic group is abelian.​

Answers

Answered by Anonymous
4

Answer:

PROOF:---

Let G be a cyclic group with a generator g∈G.

Namely, we have G=⟨g⟩ (every element in G is some power of g.)

Let a and b be arbitrary elements in G.

Then there exists n,m∈Z such that a=gn and b=gm.

It follows that

ab=gngm=gn+m=gmgn=ba.

Hence we obtain ab=ba for arbitrary a,b∈G.

Thus G is an abelian group.

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Answered by AlluringNightingale
0

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 ,

Hence ,Every cyclic group is an abelian group .

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