homework 19 (due Apr 27)

due date

Tuesday, April 27

instructions

Read Chapter 8 of our textbook up to and including Theorem 8.5. Complete the following exercises:

7, 31, 32

some hints

7. See the start of the proof of Theorem 8.3. You need to prove that the map $\phi$ is (i) 1-1, (ii) onto, and (iii) respects the group operation (i.e., $\phi(k+m)=\phi(k)\phi(m)$ for all $k,m\in \mathbb{Z}_n$).

31. Given an isomorphism $\phi$, Theorem 8.1 tells us that $\phi^{-1}$ is an isomorphism. To show that the composition of two compatible isomorphisms is an isomorphism, you need to argue that the composition is (i) 1-1, (ii) onto, and (iii) respects the group operation. But (i) and (ii) follow from known results from earlier classes. Check (iii) by hand. To show that "up to isomorphism" is an equivalence relation, you need to show that the relation is symmetric (use inverses), reflexive (easy), and transitive (use composition).

32. Both groups are cyclic. Apply Theorem 8.3 and then the previous exercise.

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