Math 360, Fall 2017, Assignment 8

Mathematical proofs, like diamonds, are hard as well as clear, and will be touched with nothing but strict reasoning.

- John Locke, Second Reply to the Bishop of Worcester

Read:

1. Section 9

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

1. Group of permutations.
2. Order (of a permutation).
3. Orbit (of a permutation).
4. Moving orbit (of a permutation).
5. Cycle.
6. Disjoint (cycles).

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

1. Cayley's Theorem.
2. Theorem concerning the order of multiplication of disjoint cycles.
3. Disjoint cycle decomposition.
4. Formula for the order of a permutation in terms of its cycle lengths.

Solve the following problems:

1. Section 9, problems 1, 3, 5, 7, 9, 11 (you may ignore the instruction concerning transpositions), 15, and 17.
2. Compute the product $(1,4)(1,3)(1,2)$ (i.e. find its disjoint cycle decomposition).
3. Compute the product $(2,7)(2,9)(2,3)(2,5)(2,6)$.
4. A cycle of length two is called a tranposition. Write the cycle $(4,2,6,9,3,1,5)$ as a product of transpositions. (The transpositions do not need to be disjoint.)
5. Show that any permutation can be written as a product of transpositions.
6. Show that $S_n$ is generated by transpositions (i.e. the smallest subgroup of $S_n$ containing all transpositions is $S_n$ itself).
7. (Symmetry group of the methane molecule) Chemists are often interested in the symmetry group of a molecule, i.e. the group of rigid motions in $\mathbb{R}^3$ that superimpose the molecule on itself. The methane molecule consists of a carbon atom surrounded by four hydrogen atoms in the shape of a regular tetrahedron; see this page for a picture. "Compute" the symmetry group $G$ of methane, as follows: first, label the hydrogen atoms with integers $1,2,3,4$. Argue that a symmetry of methane will permute the hydrogens, and will be determined by this permutation, so that $G$ is isomorphic to some subgroup of $S_4$. Draw pictures showing that all six transpositions $(1,2), (1,3), (1,4), (2,3), (2,4),$ and $(3,4)$ belong to $G$. Finally, say precisely which subgroup of $S_4$ the symmetry group is isomorphic to.
8. (Optional) Try to repeat the previous exercise for cubane.

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