homework 13 (due Mar 30)

due date

Tuesday, March 30

instructions

Read Chapter 4 of our textbook up to and including Theorem 4.6. Complete the following exercises:

3abce, 24, 26ab

some hints

For 26a, it is enough to prove that you can write any cycle as a product of $(12), (13), \ldots, (1n)$. Consider an arbitrary cycle $(a_1 a_2 \cdots a_k)\in S_n$. By what we discussed in class, we know that

(1)
\begin{align} (a_1 a_2 \cdots a_k)=(a_1 a_k)(a_1 a_{k-1})\cdots (a_1 a_2). \end{align}

The problem here is that $a_1$ may not be equal to 1. First, see if you can figure out what $(1 a_k)(1 a_{k-1})\cdots (1 a_2)$ is equal to. This isn't what you want, but it's close. The trick is to figure out what additional transpositions to multiply $(1 a_k)(1 a_{k-1})\cdots (1 a_2)$ by to get $(a_1 a_2 \cdots a_k)$ (Hint: you can do it with one on the left and one on the right).

For 26b, it is enough to prove that you can write any cycle as a product of $(12), (23), \ldots, (n-1\ n)$. First, what does 26a tell you? If you can figure out how to write any $(1m)$ as a product of $(12), (23), \ldots, (n-1\ n)$, you're done. By the way, the transpositions $(12), (23), \ldots, (n-1\ n)$ are called adjacent transpositions.

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