homework 22 (due May 6)

Thursday, May 6

# instructions

Read Chapter 9 of our textbook up to and including Example 12. Complete the following exercises:

5ac, 8, 15, 16

# some hints

5a. See back of book.

5c. See back of book.

8. See back of book.

15. First, note that $\phi(G)=\{h\in H: \exists g\in G\text{ such that } \phi(g)=h\}$ is a subgroup of $H$ by Proposition 9.3(3). To show that $\phi(G)$ is cyclic when $H$ is cyclic, you need to show that every element in $\phi(G)$ can be written in terms of a single element in $\phi(G)$. Use the image of a generator of $G$.

16. Note that "a cyclic group is completely determined by its action on the generator of the group" means that if you know the image of a generator for a group, then you know the image of every element of the group. Let $\phi:G\to H$ be a homomorphism and assume that $G$ is cyclic. Then there exists $a\in G$ such that $\langle a \rangle=G$. Let $g\in G$. What is image of $g$ under $\phi$?

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