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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