Read: Finish Section 3.3 and start 4.1.
Turn in: 3.64, 3.65, 4.1, 4.2
- For 3.64, you want to prove that \(G_2\) is abelian, so start with arbitrary \(y_1,y_2\in G_2\). To move forward, you will need to make use of the fact that \(\phi\) is onto: this says that every element \(y \in G_2\) can be written as \(y = \phi(x)\) for some \(x\in G_1\).
- The hardest part of 3.65 is to make sense of the definition $\phi(H)$. When you get to proving that $\phi(H)$ is a subgroup of \(G_2\), you should find that you only need to use that $\phi$ satisfies the homomorphic property; it won’t be necessary that $\phi$ is one-to-one and onto.