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.
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.
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
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?