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Section9.1Preparation

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.

Subsection9.1.1Videos

Subsection9.1.2Video Extras

Subsection9.1.3Homework 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 9.3, #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 9.3, #2.
  3. Exercises Section 9.3, #4.
  4. Exercises Section 9.3, #6.
  5. Exercises Section 9.3, #7.
  6. Exercises Section 9.3, #8.
  7. Exercises Section 9.3, #11.
  8. Exercises Section 9.3, #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?