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## Section9.2Preparation

###### ObjectivesGoals
• Classify a subgroup $H\lt G$ as a normal subgroup.
• Determine the factor group $G/N$ of a normal subgroup.
• Determine whether a group is an internal direct product of its subgroups.
• Classify groups of prime-square order.

### Subsection9.2.3Homework 9

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.

1. Exercises Section 9.4, #1. Hint

In part d, use this as an opportunity to acquaint yourself with the quaternion group $Q_8\text{.}$ You can find its Cayley table here: http://mathworld.wolfram.com/QuaternionGroup.html

2. Exercises Section 9.4, #2.
3. Exercises Section 9.4, #4.
4. Exercises Section 9.4, #6.
5. Exercises Section 9.4, #7.
6. Exercises Section 9.4, #8.
7. Exercises Section 9.4, #11.
8. Exercises Section 9.4, #14.
9. Consider the two subgroups of $G = U(16)$ given by \begin{equation*} H=\{1, 15\} \text{ and } K = \{ 1, 9\}. \end{equation*} Is it true that $H\cong K\text{?}$ Is it true that $G/H \cong G/K\text{?}$
10. Consider the subgroup $H = \bigl\langle (2,2) \bigr\rangle$ in the group $G = \mathbb{Z}_4\oplus \mathbb{Z}_4.$ How many elements are in the factor group $G/H\text{?}$ To what group is $G/H$ isomorphic?
11. Consider the subgroup $H = \bigl\langle (2,2) \bigr\rangle$ in the group $G = \mathbb{Z}\oplus \mathbb{Z}.$ How many elements are in the factor group $G/H\text{?}$ To what group is $G/H$ isomorphic?
12. The additive group of real numbers $\mathbb{R} = (\mathbb{R},+)$ has as a subgroup the additive group of integers $\mathbb{Z} = (\mathbb{Z},+).$ Give an example of an element of order 12 in the factor group $\mathbb{R}/\mathbb{Z}\text{.}$ Also, give an example of an element of infinite order in $\mathbb{R}/\mathbb{Z}\text{.}$ Hint

You can think of $\mathbb{R}/\mathbb{Z}$ as the "real numbers mod 1." What kinds of cosets in $\mathbb{R}$ does $\mathbb{Z}$ have? How might this problem be different with $\mathbb{Q}/\mathbb{Z}$ instead?