# due date

Tuesday, April 20

# instructions

Read Chapter 8 of our textbook up to and including Theorem 8.1. Complete the following exercises:

1, 3, 4, 27

# some hints

For 1, find an isomorphism $\phi: \mathbb{Z}\to n\mathbb{Z}$. You'll need to argue that your function is 1-1, onto, and preserves the group operation.

For 3, determine whether $U(8)$ is cyclic. What does Theorem 8.1 have to say?

On 4, it will probably be difficult to define an isomorphism in terms of a formula. If you can't think of a formula, then describe the isomorphism by saying exactly what maps to what. You can't do this in any random way, but there are a few ways to do it. Each element must map to an element of the same order, and in particular, you'll need the identity to correspond to the identity in the other group. To show that your function preserves the operation, you'll have to do 10 brute force calculations.

For 27, see Bob (Back of book).