Question

Abstract Algebra; Please write nice and clear.

If we wanted to use the definition of isomorphism to prove that Z is not isomorphic to Q, we would have to show that there does not exist an isomorphism p : Z Q. In other words, we would have to show that every function that we could possibly define from Z to Qwould violate at least one of the conditions that define isomorphisms. To show this directly seems daunting, if not impossible. The next activity provides one doable example. Activity 29.13. To show that Z is not isomorphic to Q, suppose, to the contrary, that there does exist a function p:ZQ that is both bijective and operation-preserving. Since 1 generates Z, we will consider the rational number q - p(1 a) Explain why p(n)nq for every integer n. (b) Is p an injection? Prove your answer. (c) Is pa surjection? Prove your answer. (d) Deduce that Z cannot be isomorphic to Q. (While it may seem difficult to believe, there are bijections from Z to Q, but they cannot preserve the structure of Z.)

Answer 1

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