Math 360, Fall 2018, Assignment 12

"When I think of Euclid even now, I have to wipe my sweaty brow."

- C. M. Bellman

Read:

1. Section 9.

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

1. Sign (of a permutation).
2. $A_n$ (the alternating group on $n$ letters).

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

1. Rule concerning the parity of a cycle.
2. Rules concerning the parity of a product.
3. Theorem concerning the sign of the product.
4. Theorem relating the number of even permutations to the number of odd permutations.
5. Formula for the order of $A_n$.

Solve the following problems:

1. Section 9, problems 10, 11, and 12 (in a previous assignment, you have already expressed each permutation as a product of cycles; now express each as a product of transpositions, and compute its parity).

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