If G is a group that (ab)^n = a^n.b^n , for all a, b ϵG and for three consecutive integers n. Then prove that G is an abelian group.

If  G is a group that (ab)n = an.bn , for all a, b ϵG and for three consecutive integers n. Then prove that G is an abelian group.

Proof:

Let a and b be any two elements of G.

Suppose n, n+1 , n + e are three consecutive integers such that

(ab)n = an.bn …………….(1)

(ab)n+1 = an+1.bn+1 ………..(2)

(ab)n+2 = an+2. bn+2……………(3)

Equation (2) can also be written as,

(ab)(ab)n = a.an.b.bn

a.b.an.bn = a.an.b.bn

b.an = an.b [by cancellation law]

Equation (3) can also be written as,

(ab)(ab)n+1 = a.an+1.b.bn+1

a.b.an+1.bn+1 = a.an+1.b.bn+1

b.an+1 = an+1.b [by cancellation law]

b.an.a = an.a.b

an.b.a = an.a.b

⇒ ba = ab [by left cancellation law]

Therefore, G is an abelian group.

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