Math 120A HW 7

1. Let $n \in \mathbb{Z}^+$ and let $\sigma \in S_n$. Let $B_1,\ldots,B_r$ denote the non-singleton orbits of $\sigma$. For $i \in \{1,\ldots,r\}$, define the permutation $\sigma_i \in S_n$ by \[\sigma_i(x) = \begin{cases} \sigma(x) & \text{if }x \in B_i\\ x & \text{if }x \notin B_i. \end{cases}\] Recall that each such permutation $\sigma_i$ is a cycle with support $B_i$, and the supports $B_1,\ldots,B_r$ are pairwise disjoint. Prove that the product $\sigma_1\cdots \sigma_r$ is equal to $\sigma$ (this was the remaining step in our proof that every permutation is a product of disjoint cycles; the book says it is "clearly" true, but let's take a moment to prove it.) Hint: consider an arbitrary element $x \in \{1,\ldots,n\}$. We want to show that $\sigma_1\cdots \sigma_r(x) = \sigma(x)$. Consider cases. In some cases, it helps to break up the product $\sigma_1\cdots \sigma_r$ into smaller products and then say how each smaller product moves the relevant element.
2. Consider the product of transpositions $\sigma = (2\;5)(3\;5)(2\;4)(2\;3)$ in $S_5$.
   1. Express $\sigma$ as a product of disjoint cycles.
   2. Express $\sigma$ as a product of transpositions, using as few transpositions as possible.
   3. Can $\sigma$ be expressed as a product of disjoint transpositions? If so, do it. If not, say why not.
3. List all the elements of the alternating group $A_4$ using cycle notation (i.e. write every non-identity element of $A_4$ as a product of disjoint cycles.) Note that $\left|A_4\right| = 4!/2 = 12$.
4. In this problem we give geometric interpretations of the alternating groups $A_3$ and $A_4$.
   1. Draw an equilateral triangle and label its vertices $1$, $2$, and $3$ in any order. Every element of $S_3$ corresponds to a symmetry of the triangle (as we have discussed.) What do the elements of $A_3$ correspond to, geometrically speaking?
   2. Draw a square and label its vertices $1$, $2$, $3$, and $4$ in clockwise or counterclockwise order. Does every element of $A_4$ correspond to some symmetry of the square in the same way that every element of $A_3$ corresponds to some symmetry of the equilateral triangle? If so, why? If not, why not?
   3. Draw a regular tetrahedron and label its vertices $1$, $2$, $3$, and $4$ in any order. Every element of $S_4$ corresponds to some symmetry of the tetrahedron (you do not need to prove this.) For each element of $A_4$, say what it corresponds to, geometrically speaking.

      For example, letting $\ell_1$ denote the line passing through the vertex $1$ and the center of the opposite face $\triangle 234$, the elements $(2\; 3\; 4)$ and $(2\;4\;3)$ of $A_4$ correspond to rotations by $1/3$ turn in opposite directions around the line $\ell_1$.