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HW2_soln - Homework#2 Solutions 14 Suppose that G is a...

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Homework #2 Solutions 14. Suppose that G is a group with the following property: for any a, b, c G , ab = ca implies b = c . Let x, y G . Set a = x - 1 , b = xy and c = yx . Then ab = x - 1 ( xy ) = ( x - 1 x ) y = ey = y = ye = y ( xx - 1 ) = ( yx ) x - 1 = ca. By our hypothesis, we must have xy = b = c = yx . Since x and y were arbitrary, we conclude that G is abelian. 16. Let G be a group and let a, b G . Using the associativity property of groups we have ( ab )( b - 1 a - 1 ) = a ( bb - 1 ) a - 1 = aea - 1 = aa - 1 = e and ( b - 1 a - 1 )( ab ) = b ( aa - 1 ) b - 1 = beb - 1 = bb - 1 = e. Since inverses are unique, we must have ( ab ) - 1 = b - 1 a - 1 . Note: In class I showed that any one-sided inverse in a group is automatically a two-sided inverse. Therefore, any one of the above inequalities also establishes the result. 20. We will prove by induction that if G is a group, n Z + and a 1 , a 2 , . . . , a n G then ( a 1 a 2 · · · a n ) - 1 = a - 1 n a - 1 n - 1 · · · a - 1 2 a - 1 1 . There is nothing to prove if n = 1. So, assume that the result holds for some n 1. Let a 1 , a 2 , . . . , a n +1 G . Then, according to the previous problem and the inductive hypothesis
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