Read: Continue with Section 3.3.
Turn in: 3.43, 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, read on: 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.
For 3.60, 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 $m\in \mathbb{Z}$ 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, intuitively, 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 for $G_2$ is $\phi(a)$.
Extra practice: 3.59
