Math 360, Fall 2014, 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. Group of permutations.
2. Dihedral group.
3. Orbits (of a permutation).
4. Cycle.
5. Disjoint cycles.
6. Transposition.
7. Even permutation.
8. Odd permutation.
9. Alternating group (on $n$ letters).

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

1. Theorem concerning the order of the dihedral group.
2. Cayley's Theorem.
3. Theorem concerning generation of $S_n$ by transpositions (Corollary 9.12).
4. Theorem concerning the parity of a permutation (Theorem 9.15).
5. Theorem concerning the order of the alternating group (Theorem 9.20).

Solve the following problems:

1. Section 8, problems 25 and 26.
2. Section 9, problems 3, 5, 9, 10, 15, and 24.

--------------------End of assignment--------------------