If xy x 1y 1 for all x and y in G prove that G must be abel

If xy = x 1y 1 for all x and y in G, prove that G must be abelian.

Solution

Let G be a group and for all x , y in G.

Suppose xy = x^-1 y^-1 for all x , y in G ----------------- ( 1 )

Now we show that xy = yx for all x , y in G ;

But if  x , y in G then (xy)^-1 = y^-1x^-1.

So that consider (xy)^-1 = y^-1x^-1. = x^-1 y^-1 by ( 1 )

= > x^-1 = y x^-1 y^-1

= > e = xyx^-1 y^-1

= > y = xyx^-1

= > yx = xy

Thus for all x , y in G we proved that xy = yx . Hence G is abelian