Math 360, Fall 2019, 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. $\mathrm{sgn}(\pi)$ (the sign of the permutation $pi$).
2. Even permutation.
3. Odd permutation.
4. $A_n$ (the alternating group on $\{1,\dots,n\}$).

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

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

Solve the following problems:

1. Section 9, problem 29.
2. Section 15, problem 39(b).

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