Read: Section 7.1.
Turn in: 7.16(a,b), 7.18, 7.19, 7.20
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On 7.16, the idea is that if you know that $\phi(i)= h$ and $\phi(j) = v$, then, for example, you can figure out $\phi(k) = \phi(ij) = \phi(i)\phi(j)=hv$.
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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.
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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.