Prove that a group G is abelian iff a1b1 ab1 for all a b el

Prove that a group G is abelian iff a^-1b^-1 = (ab)^-1 for all a, b elementof G.

Solution

Let G be Abelian, that is ab = ba, for any a,b G,

Then a¹b¹. = (ba)^-1 = (ab)^-1

Now, Let assume (ab)¹  = a¹b¹ for all a,b G.

Then (ab)(ab)¹ = e and (ba)(ab)1 = ba(a1b1) = e.

By cancellation, ab = ba.