Math 360, Fall 2015, Assignment 7

Mathematicians are like Frenchmen: whatever you say to them they translate into their own language, and forthwith it is something entirely different.

- Goethe

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

1. Orbit (of a permutation).
2. Cycle.
3. Disjoint (cycles).

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

1. Cayley's theorem.
2. Theorem concerning the order in which disjoint cycles are multiplied.
3. Theorem concerning disjoint cycle decomposition of elements of $S_n$.
4. Theorem concerning the order of an element of $S_n$.

Solve the following problems:

1. Section 8, problems 7 and 9.
2. Section 9, problems 1, 3, 5, 7, 9, 15, and 17. (Note: in problems 7 and 9, the expression "compute the product" means "find the disjoint cycle decomposition of the product.")
3. Find an isomorphism from the Klein four-group $V$ to some group of permutations. (Hint: follow the proof of Cayley's theorem, letting $V$ permute itself by multiplication. This will embed $V$ as a subgroup of $\mathrm{Sym}(V)=\mathrm{Sym}(\{e, a, b, c\})\simeq S_4.$ The copy of $V$ inside $S_4$ plays a major role in Cardano's formula for the solution of the general quartic equation, and the absence of any analogous subgroup of $S_5$ is related to the unsolvability of the general quintic by radicals.)

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