Homework 26

Read: Read Section 7.2.

Turn in: 7.18, 7.19, 7.22, 7.23, 7.24, 7.25

• For 7.18, it might be good to first limit the possibilities for homomorphisms from $\mathbb{Z}_3$ to $\mathbb{Z}_6$; then from the limited list, you can decide which are actually homomorphisms. To do this, assume that $\phi: \mathbb{Z}_3 \rightarrow \mathbb{Z}_6$ is a homomorphism. What are the possible values for $\phi(0)$, $\phi(1)$, and $\phi(2)$? By the homomorphic property, once you know $\phi(1)$, then you know $\phi(0)$ and $\phi(2)$ since $1$ is a generator for $\mathbb{Z}_3$. So the question is: what are the possible values for $\phi(1)$? Each possibility gives you a different possible homomorphism $\phi$, but notice how Theorem 7.7 limits the possibilities.

• For 7.19, start by assuming that $\phi: D_3 \rightarrow \mathbb{Z}_3$ is a homomorphism. Your goal is to show that $\phi(g) = 0$ for all $g\in D_3$, and by the homomorphic property, it suffices to just show that $\phi(r) = 0$ and $\phi(s) = 0$ since $\{r,s\}$ is a generating set for $D_3$. Now you want to consider the possible values for $\phi(r)$ and $\phi(s)$, and these are limited by Theorem 7.7.

• For 7.22–7.25, you are trying to show $G_1/N\cong G_2$ for different choices of $G_1$, $N$, and $G_2$. This can be done using the First Isomorphism Theorem by following these steps: (1) define a homomorphism $\phi:G_1 \rightarrow G_2$ (this can be the hardest step…unless it’s already done for you), (2) show that $\phi(G_1) = G_2$, and (3) show that $\ker(\phi) = N$. Once you’ve done that, the First Isomorphism Theorem tells you that $G_1/N\cong G_2$.

  • In 7.23, you are given a function $\phi$, but you still need to justify that it is a homomorphism.
  • Problems 7.3 and 7.9 are helpful for 7.25.

Extra practice: 7.26, 7.28