Math 360, Fall 2013, Assignment 6

I tell them that if they will occupy themselves with the study of mathematics, they will find in it the best remedy against the lusts of the flesh.

- Thomas Mann, The Magic Mountain

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

1. Orbits (of a permutation).
2. Cycle.
3. Disjoint cycles.
4. Transposition.
5. Even permutation.
6. Odd permutation.
7. Alternating group on $n$ letters.

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

1. Theorem concerning generation of $S_n$ by transpositions (Corollary 9.12 in the text).
2. Theorem concerning the parity of a permutation (Theorem 9.15).
3. Theorem concerning the order of the alternating group (Theorem 9.20).

Solve the following problems:

1. Section 9, problems 3, 5, 9, 10, 15, 24, and 29.

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