Hint to puzzle 129: Abelian group

For any two group elements x, y:

Consider ((xy)⁻¹(xy)²(yx)⁻¹)² = ((xy)⁻¹(yx)²(yx)⁻¹)².

-- or --

1. (i)    By considering ((xy⁻¹)y)²y, show that x² and y commute;
2. (ii)   Show that x⁻¹y⁻¹x = xy⁻¹x⁻¹;
3. (iii)  Show that (xyx⁻¹y⁻¹)² = e, the identity element.