Math 120A HW 6

1. Let $A$ be a set. For a permutation $\sigma$ of $A$, we define the support of $\sigma$ by $\text{supp}(\sigma) = \{x \in A : \sigma(x) \ne x\}$ (so it is the set of everything that is moved by $\sigma$.) Say that two permutations $\sigma,\tau \in S_A$ are disjoint if their supports are disjoint sets: $\text{supp}(\sigma) \cap \text{supp}(\tau) = \emptyset$. Prove that if $\sigma,\tau \in S_A$ are disjoint permutations, then they commute: $\sigma\tau = \tau \sigma$.
2. Let $n$ be an integer greater than or equal to $3$. Recall that the dihedral group $D_n$ is defined as $\{\sigma_1,\ldots,\sigma_n\} \cup \{\rho_1,\ldots,\rho_n\}$ where the rotation $\sigma_k \in S_n$ is defined by $\sigma_k(i) \equiv k + i \pmod{n}$ and the reflection $\rho_k \in S_n$ is defined by $\rho_k(i) \equiv k - i \pmod{n}$. Prove that $D_n$ is closed under composition. Hint: there are four cases to consider. The composition of a rotation and a rotation is a rotation (which one?) et cetera.
3. List the elements of the symmetric group $S_3$, and for each one, list its orbits.
4. In this problem, we go the other way: from orbits to permutations.
   1. List all elements of the symmetric group $S_4$ that have two orbits of cardinality 2.
   2. List all elements of the symmetric group $S_4$ that have one orbit of cardinality 3 and another of cardinality 1.
   Hint: First consider the possible ways to partition the set $\{1,2,3,4\}$ into orbits with the desired property, and then consider the possible ways to define permutations with such orbits.
5. In $S_6$, consider the following product of cycles: $\sigma = (2\;6\;1)(3\;6)(1\;2\;4)$.
   1. Compute $\sigma$ (write it in the two-row permutation notation.)
   2. Write $\sigma$ as a product of disjoint cycles.
6. Write each element of the dihedral group $D_6$ as a product of disjoint cycles. (For example, $\sigma_3 = (1\;4)(2\;5)(3\;6)$.)