Homework 17

Read: Continue with Section 4.1.

Turn in: 4.29, 4.32, 4.36, 4.37, 4.38, 4.40, 4.43

For 4.29 you will want to construct an isomorphism between $G$ and $\mathbb{Z}$. I recommend defining it from $\mathbb{Z}$ to $G$. You know $G$ is cyclic, so $G = \langle g \rangle = \{g^k \mid k \in \mathbb{Z}\}$. To define $\phi$, you need to decide how to complete this: $\phi(n) = g^{??}$. Once you’ve done this, you need to verify that the $\phi$ you defined is one-to-one, onto, and a homomorphism. You will need 4.28; do you see why?
As you start working with $\mathbb{Z}_n$ and $U_n$, remember that the operation is addition for $\mathbb{Z}_n$ but multiplication for $U_n$. Also, one small trick for working modulo $n$ is that you can always add or subtract $n$ without changing the object. For example, $5,7\in U_{12}$, so $5\cdot 7 \equiv 35 \equiv 35 - 12 - 12 \equiv 11$ (mod $12$). Thus, in $U_{12}$, $5\cdot 7 = 11$. Similarly, in $\mathbb{Z}_{12}$, $5+7 \equiv 12 \equiv 12-12 \equiv 0$ (mod $12$), so $5+7 = 0$ in $\mathbb{Z}_{12}$. Feel free to look at other resources (e.g. https://en.wikipedia.org/wiki/Modular_arithmetic) to fill in background about modular arithmetic if needed.
Regarding notation, note that $G\cong H$ means “$G$ is isomorphic to $H$”.
On 4.40, you just need to properly link together Theorems 4.27, 4.29, and 4.39. Also, remember that Theorem 3.54 implies that if $G_1 \cong G_2$ and $G_2 \cong G_3$, then $G_1 \cong G_3$.
Your proof of 4.43 might be quite short; you want to make use of Theorem 4.42.

Extra practice: 4.28, 4.34, 4.35, 4.39, 4.41, 4.42

4.41 is a bit hard. You can write $G = \langle g \rangle$, and then your first task it to try to guess a generator for $H$. Spoiler alert!! …for a generator of $H$, try choosing $a = g^m$ where $m$ is the smallest positive integer such that $g^m \in H$. Now let $h \in H$; you want to show that $h$ is a power of $a$. you can write $h = g^n$ for some $n\in \mathbb{Z}$, and now try dividing $n$ by $m$ and see if you can prove that the remainder is $0$.