Prove that if (ab)^-1 = a^-1b^-1, for all a, b ϵG then G is abelian

Prove that if (ab)-1 = a-1b-1, for all a, b ϵG then G is abelian

Proof:

Given (ab)-1 = a-1.b-1…………………..(1), for all a, b ϵG

Since G is a group, we have (ab)-1 = b-1.a-1 ………….(2), for all a, b ϵ G

From (1) and (2), b-1a-1 = a-1b-1

Taking the inverse

(b-1a-1)-1 = (a-1 b-1)-1

(b-1)-1(a-1)-1 = (a‑1)-1(b-1)-1 [since (a-1)-1 = a, for all a ϵG]

⇒ba = ab

⸫ G is an abelian group.

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One thought on “Prove that if (ab)^-1 = a^-1b^-1, for all a, b ϵG then G is abelian

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