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Unformatted text preview: e . 10. Suppose that for every a G , a = a-1 . Then a 2 = e . Let a and b G , and set c = ab . Then e = c 2 = ( ab ) 2 and a 2 b 2 = e e = e. Thus the equation ( ab ) 2 = a 2 b 2 , is satised for somewhat vacuous reasons. Now apply (3). 4. Suppose that ( ab ) i = a i b i and ( ab ) i +1 = a i +1 b i +1 . Expanding the second equality, we get a ( ba ) i b = aa i b i b. 1 Multiplying on the left by a-1 and by b-1 on the right, we get ( ba ) i = a i b i = ( ab ) i . Hence we have c = ( ba ) i = ( ab ) i and ( ba ) i +1 = ( ab ) i +1 . Multiplying both sides of the second equation by c-1 , we get ab = ba, which is what we want. 2...
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