jan20 - Math 4124 Wednesday, January 20 January 20,...

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Math 4124 Wednesday, January 20 January 20, Ungraded Homework Exercise 1.1.5 on page 21 Prove for all n > 1 that Z / n Z is not a group under multiplica- tion of residue classes. Z / n Z = { ¯ 0 ,..., n - 1 } . Suppose Z / n Z is a group. Then it must have an identity e . We now have e = e ¯ 1 = ¯ 1, so the identity is ¯ 1. Let x be the inverse for ¯ 0. Then e = x ¯ 0 = ¯ 0. We conclude that ¯ 1 = ¯ 0, which is a contradiction, unless n = 1. Exercise 1.1.8 on page 22 Let G = { z C | z n = 1 for some n Z + } . (a) Prove that G is a group under multiplication (called the group of roots of unity in C ). (b) Prove that G is not a group under addition. (a) Note that multiplication defines a binary operation on G , because if x , y G , then x m = 1 and y n = 1 for some m , n Z + , and then we have ( xy ) mn = 1, which shows that xy G . Also multiplication is associative and the identity is 1; note that 1 G . Finally if z G , then it has an inverse z - 1 ; again note that z - 1 G because if
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