Lecture 9

Navigation

• Back to Main Page.
• Back to the Lectures.
• Before the Lecture
• Related Problems
• In Class
• Outcomes

Before the Lecture

• Quickly read Section 2.1 (review on functions, including one-to-one and onto), if you haven't done it already.
• Read Section 2.2:
  • Focus on computations!
  • Permutations (matrix notation, computing composition and inverses, inverse of a composition). Careful: Like functions, read compositions from right to left!
  • Cycles.
  • Decomposition into disjoint cycles/complete factorization (Properties 2.24, Theorem 2.26).
  • Conjugation of permutations (Lemma 2.31 and Proposition 2.32).
  • Decomposition as composition of transpositions (Proposition 2.35)
  • You may skip Example 2.36 and Lemmas 2.37 and 2.38. (They are important to the proof of the formulas for sign below.)
  • Sign (Definition, Theorems 2.39 and 2.40 and Corollary 2.41).
  • Skip Example 2.42.
• Watch the videos related to this section (after reading it):
  • A few words on permutations.
  • Computations with Permutations.
  • Permutations and Determinant.
• Bring questions from previous sections too, as this meeting is also planned for a quick catch up.
• Write down all questions about the above topics to bring to our (online) lecture. (You can also type them in the file "Questions.tex" in SageMathCloud.) Comments about the videos are welcome!
• Work on the assigned problems for these sections. (See Related Problems below.) You don't need to finish them, but try to work on as many as you can and the bring your questions to class.

 

Related Problems

The "turn in" problems are due on 06/29 by 11:59pm.

Section 2.2:	Turn in: 2.21(x), 2.34 (note that $\alpha \beta = \beta \alpha$ iff $\alpha \beta \alpha^{-1} = \beta$) and this question on permutations.
	Extra Problems: 2.21 ((ii) is easier after 2.3), 2.22, 2.25, 2.26, 2.34, this question on permutations.

 

In Class

In class:

• We will discuss the reading and pace.
• I will discuss the main points.
• I will answer questions about the sections covered.
• I will answer any other questions.
• We can work on the HW problems.

 

Outcomes

After the assignment (reading and videos before class) and class, you should:

• know the definition and basic properties of permutations;
• know that permutations in general do not commute, but disjoint cycles do;
• be able to compute compositions, inverses, factorization into disjoint cycles, factorization into transpositions and sign of permutations (Problem 2.22 and FIX!! modified old exam problem.);
• understand how conjugation (the conjugation of $\alpha$ by $\beta$ is $\beta \alpha \beta^{-1}$) affects cycles (and products of cycles) and be able to apply this idea.