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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.