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Section4.1Preparation

ObjectivesLearning Goals for this Chapter

ObjectivesKey Definitions and Theorems

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...