
## Section4.1Preparation

### Subsection4.1.1Acquaint 6

1. Watch the videos above, and read Section 4.2 and leave comments, questions, and discussions using your http://hypothes.is account.
2. Complete Acquaint 6 by 10:00am on Wednesday 10/2.

### Subsection4.1.2Acquaint 7

1. Watch the videos above, and read Section 4.2 and leave comments, questions, and discussions using your http://hypothes.is account.
2. Complete Acquaint 7 by 10:00am on Friday 10/4.

### Subsection4.1.3Homework 4

Homework exercises are valuable preparation for your quiz on this material. Generally I do not collect homework for grading -- however, I will expect to review your homework progress in the event of your requesting a revision on a quiz. You are welcome to discuss homework problems with your groups and with the class, both in person and in the Slack #math channel.

#### Subsubsection4.1.1Exercises

1. Exercises 4.4, #1.
2. Exercises 4.4, #2.
3. Exercises 4.4, #3.
4. Exercises 4.4, #8.
5. Exercises 4.4, #11.
6. Exercises 4.4, #13.
7. Exercises 4.4, #23.
8. Exercises 4.4, #24.
9. Exercises 4.4, #26.
10. Exercises 4.4, #27.
11. Exercises 4.4, #30.
12. Let $G$ be a group and suppose that $s,t\in G$ are two involutions (elements of order two). Prove that $s$ and $t$ commute (i.e., that $st = ts$).
13. Let $G$ be a group and suppose that $s,t\in G$ are two involutions (elements of order two). Prove that either $s = t\text{,}$ or there exists a third involution in $G$ distinct from both $s$ and $t\text{.}$
14. Complete and prove the following conjecture: Let $G$ be a cyclic group. If there exists an element $g\in G$ with $|g| = \infty\text{,}$ then $G$ has ___ elements of finite order.
15. Consider the additive group of integers, $\mathbb{Z}=(\mathbb{Z},+)\text{.}$ Prove that every subgroup of $\mathbb{Z}$ is cyclic.
16. Suppose that $n = 2^k$ for some $k\geq 3\text{.}$ Prove that the multiplicative group $\mathcal{U}(n)$ is not cyclic.
17. Let $G$ be a group and $a\in G$ have infinite order. If $H = \langle a \rangle\text{,}$ describe all subgroups $K \lt H\text{.}$ Hint

The previous problem --- about the group of integers --- can provide some inspiration.

18. Sketch the subgroup lattices for the additive groups $\mathbb{Z}_8$ and $\mathbb{Z}_{16}\text{.}$
19. Suppose $G$ is a group and $a,b\in G$ are elements that commute with one another ( $ab = ba$ ). Prove that if $|a| = k \lt \infty$ is finite and $|b| = \infty\text{,}$ then $\bigl|ab \bigr| = \infty\text{.}$
20. Suppose $G$ is a group and $a,b\in G$ are elements that commute with one another ( $ab = ba$ ). Complete and prove a conjecture: If $|a| = k \lt\infty$ and $|b| = \ell \lt\infty$ are both finite, then $\bigl|ab\bigr|$ can be...