Homework 14

Read: Continue with Section 3.3. Start with Section 4.1.

Turn in: 3.60 or 3.62 (you choose one), 3.63 or 3.64 (you choose one), 4.1, 4.2, 4.4(a,b,e)

• You should only turn in 5 problems. You get to choose between 3.60 and 3.62, and you can also choose between 3.63 and 3.64. But we will discuss all of the problems.

• For both 3.60 and 3.62, you will 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: 3.61, 3.65, 3.66, 4.3