Math 360, Fall 2016, Assignment 8

Do not imagine that mathematics is hard and crabbed and repulsive to commmon sense. It is merely the etherealization of common sense.

- Lord Kelvin

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

1. Orbit (of a point under a permutation).
2. Cycle.
3. Disjoint permutations.
4. Transposition (or swap).

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

1. Disjoint cycle decomposition.
2. Theorem concerning the order of multiplication of disjoint permutations.
3. Formula for the order of a permutation (in terms of its cycle lengths).
4. Theorem concerning generation of $S_n$ by cycles.
5. Theorem concerning generation of $S_n$ by transpositions.

Solve the following problems:

1. Section 9, problems 1, 3, 5, 7, 9, 11, 15, and 17.

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