Read: Continue with Section 3.3.
Problems to Try: 3.50, 3.52, 3.53, 3.55, 3.60, 3.62
- For 3.50 you are given a function and need to check three things: (1) does it satisfy the homomorphic property in Definition 3.48, (2) is it one-to-one, and (3) is it onto. If any one of them fails, it is not an isomorphism.
- On 3.52 you need to first come up with a function from $\mathbb{Z}$ to $2\mathbb{Z}$, and then verify the three things listed above for 3.50.
- There are multiple ways to approach 3.53. If you want a hint, you could consider the fact that \(\phi(e_1) = \phi(e_1*e_1)\).
- For 3.55, remember that since \(\phi:G_1 \rightarrow G_2\) is a bijection, \(\phi^{-1}: G_2\rightarrow G_1\) is a function and is also a bijection. You do not need to prove this. Also, remember the relationship between a function and its inverse: $\phi^{-1}(\phi(x)) = x$ and $\phi(\phi^{-1}(y)) = y$. To prove that \(\phi^{-1}: G_2\rightarrow G_1\) has the homomorphic property, you should start by considering \(\phi^{-1}(y_1+y_2)\) for \(y_1,y_2 \in G_2\). You will need to find a way to rewrite \(y_1\) and \(y_2\) to move forward.
- For 3.62, you should assume that \(G_1\) is cyclic, so it is generated by some \(a\in G_1\) (i.e. \(G_1 = \langle a \rangle\)). This means that for all \(g\in G_1\) there exists a $k\in \mathbb{Z}$ such that $g = a^k$. Now, to show that \(G_2\) is cyclic, you need to find a generator $b$ for \(G_2\) so that for all \(h\in G_2\) there exists a $k\in \mathbb{Z}$ such that $h = b^k$. Your first task is to try to guess what $b$ might be. (Stop reading, and really take a guess!) Remember that an isomorphism transfers information from one group to another, so if $a$ is a generator for \(G_1\), then a good guess for a generator of \(G_2\) is $\phi(a)$. Note that you will also need to make use of the fact that $\phi$ is onto; this implies that every element \(y \in G_2\) can be written as $y = \phi(x)$ for some \(x\in G_1\). Carefully write down what you want to prove about $G_2$, and then use use that $\phi$ is onto to translate what you want to prove.