# HW1_soln - Homework#1 Solutions p 241#2 The identity...

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Unformatted text preview: Homework #1 Solutions p 241, #2 The identity element is easily seen to be 6. Indeed, in Z 10 we have 2 ยท 6 = 12 = 2 4 ยท 6 = 24 = 4 6 ยท 6 = 36 = 6 8 ยท 6 = 48 = 8 . p 241, #4 There are many possible examples. Probably the simplest occurs in Z 4 , where both 1 and 3 are solutions to 2 x = 2. We know that such a situation cannot happen in a group, for in that case the equation ax = b has the unique solution x = a- 1 b . p 241, #14 We prove the result for all nonnegative m first. If m = 0 the result is obvious. Now assume that m โฅ 1. Then m ยท ( ab ) = ab + ab + ยทยทยท + ab | {z } m times = ( a + a + ยทยทยท + a ) | {z } m times b = ( m ยท a ) b. If we had instead factored a out on the right side, we would have obtained instead m ยท ( ab ) = a ( m ยท b ). Thus, m ยท ( ab ) = ( m ยท a ) b = a ( m ยท b ) for all m โ Z + . If m < 0 then m =- n for some n > 0. We then have, using part 2 of Theorem 12.1 and the preceding result m ยท ( ab ) = (- n ) ยท ( ab ) = n ยท (- ( ab )) = n ยท ((- a...
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HW1_soln - Homework#1 Solutions p 241#2 The identity...

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