Math 120A HW 5

1. In this problem we consider permutations in $S_7$.
   1. Find the orbits of the permutation $\sigma = \begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 & 7\\ 5 & 1 & 4 & 7 & 2 & 6 & 3 \end{pmatrix}.$
   2. Write down a permutation $\tau$ whose orbits are $\{1,3\}$, $\{6\}$ and $\{2,4,5,7\}$.
   3. Write down another permutation in $S_7$ that is different from $\tau$ but has the same orbits: $\{1,3\}$, $\{6\}$ and $\{2,4,5,7\}$.
2. Let $n \in \mathbb{Z}^+$ and consider an arbitrary permutation $\sigma \in S_n$.
   1. Must $\sigma^2$ have the same orbits as $\sigma$?
   2. Must $\sigma^{-1}$ have the same orbits as $\sigma$?
   For each part of the problem, prove the statement or else find a counterexample.
3. How many elements of $S_4$ are not cycles? List them. (Hint: consider how the set $\{1,2,3,4\}$ can be partitioned into orbits.) Show that every element of $S_4$ that is not a cycle is equal to the product of two disjoint cycles. (Later we will see more generally that every element of $S_n$ can be written as the product of some number of disjoint cycles. But here you can show it explicitly.)
4. Consider the permutation $\sigma = \begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6\\ 6 & 2 & 5 & 3 & 4 & 1 \end{pmatrix}.$
   1. Write $\sigma$ as a product of disjoint cycles.
   2. Write $\sigma$ as a product of transpositions.
   (For both parts, you can check your work by checking how the permutation in your answer moves elements $x \in \{1,2,3,4,5,6\}$.)
5. Consider the permutation $\sigma = (1\;4)(3\;4)(1\;2)(1\;3)$ in $S_5$.
   1. Show that $\sigma$ is equal to a product of two transpositions.
   2. Can $\sigma$ be equal to a product of three transpositions? Why or why not?
6. Consider permutations in $S_n$. We proved a lemma in class saying that the identity permutation (which is even) is not odd (meaning it cannot be written as a product of an odd number of transpositions.) Use this lemma to prove that no permutation can be both even and odd.

Optional practice problems from the book (these will not be graded): Section 9 Exercises 1, 5, 7, 11, 13d, 21, 23