Math 360, Fall 2020, Assignment 11

I have found a very great number of exceedingly beautiful theorems."

- Pierre de Fermat

Carefully define the following terms, then give one example and one non-example of each:

1. $\mathrm{orb}(\sigma)$ (the orbit count of the permutation $\sigma$).
2. $\mathrm{sgn}(\sigma)$ (the sign of the permutation $\sigma$).
3. Even permutation.
4. Odd permutation.
5. $A_n$ (the alternating group on $\{1,\dots,n\}$).

Carefully state the following theorems (you do not need to prove them):

1. Theorem relating $\mathrm{orb}(\tau\sigma)$ to $\mathrm{orb}(\sigma)$ (where $\tau$ is a transposition).
2. Theorem relating $\mathrm{sgn}(\tau\sigma)$ to $\mathrm{sgn}(\sigma)$ (where $\tau$ is a transposition).
3. Theorem relating $\mathrm{sgn}(\sigma)$ to the number of transpositions in any expression of $\sigma$ as a product of transpositions.
4. Theorem relating $\mathrm{sgn}(\pi\sigma)$ to $\mathrm{sgn}(\pi)$ and $\mathrm{sgn}(\sigma)$ (where $\pi$ and $\sigma$ are arbitrary permutations).
5. Formula for the order of $A_n$.

Solve the following problems:

1. Let $B_n$ denote the set of odd permutations in $S_n$. (We noted in class that $B_n$ is not a subgroup of $S_n$, but it is a well-defined subset of $S_n$.) Show that when $n\geq2$, the map $f:A_n\rightarrow B_n$ defined by $f(\sigma)=(12)\sigma$ is a bijection.
2. Prove that the order of $A_n$ is $n!/2$.
3. Section 9, problem 29. (Hint: follow the technique of the previous two problems.)
4. Explicitly describe (i.e. make an operation table for) the group $A_3$. Can you find a more familiar group that is isomorphic to $A_3$?
5. For which $n$ is $A_n$ an abelian group? Justify your answer.
6. Section 15, problem 39(b).

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