Math 120A HW 5
Due Tuesday, November 18.
- In this problem we consider permutations in $S_7$.
- Find the orbits of the permutation
$\sigma = \begin{pmatrix}
1 & 2 & 3 & 4 & 5 & 6 & 7\\
5 & 1 & 4 & 7 & 2 & 6 & 3
\end{pmatrix}.$
- Write down a permutation $\tau$ whose orbits are $\{1,3\}$, $\{6\}$ and $\{2,4,5,7\}$.
- Write down another permutation in $S_7$ that is different from $\tau$ but has the same orbits: $\{1,3\}$, $\{6\}$ and $\{2,4,5,7\}$.
- Let $n \in \mathbb{Z}^+$ and consider an arbitrary permutation $\sigma \in S_n$.
- Must $\sigma^2$ have the same orbits as $\sigma$?
- Must $\sigma^{-1}$ have the same orbits as $\sigma$?
For each part of the problem, prove the statement or else find a counterexample.
- 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.)
- Consider the permutation $\sigma = \begin{pmatrix}
1 & 2 & 3 & 4 & 5 & 6\\
6 & 2 & 5 & 3 & 4 & 1
\end{pmatrix}.$
- Write $\sigma$ as a product of disjoint cycles.
- 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 $x \in \{1,2,3,4,5,6\}$.)
- Consider the permutation $\sigma = (1\;4)(3\;4)(1\;2)(1\;3)$ in $S_5$.
- Show that $\sigma$ is equal to a product of two transpositions.
- Can $\sigma$ be equal to a product of three transpositions? Why or why not?
- 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.
There are only 6 problems assigned this week
Optional practice problems from the book (these will not be graded): Section 9 Exercises 1, 5, 7, 11, 13d, 21, 23