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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Questions:

Exercise 9.15:

In this exercise, we are supposed to find the maximum possible order for an element of $S_6$. The book states that the answer is 6, but shouldn't the answer be 8? If we have the product of disjoint cycles of length 2 and 4, the greatest order would be 8. --Sean.Fraser (talk) 10:40, 24 October 2015 (EDT)

The order of a permutation is the least common multiple of the cycle lengths, not the product. (If the cycles have length 2 and 4, then on every fourth turn of the wheel, both cycles will have returned to their starting positions. So the order will be 4, not 8.) --Steven.Jackson (talk) 21:42, 25 October 2015 (EDT)