Read: Section 7.1.
Turn in: 7.22, 7.23, 7.24, 7.25
- 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 briefly justify that it is a homomorphism.
- Problem 7.3 can be helpful for 7.25 when trying to find a homomorphism from \(\mathbb{Z}_4 \times \mathbb{Z}_2\) to \(\mathbb{Z}_4\).
Extra practice: 7.26, 7.28