Math 360, Fall 2019, Assignment 4

I was at the mathematical school, where the master taught his pupils after a method scarce imaginable to us in Europe. The proposition and demonstration were fairly written on a thin wafer, with ink composed of a cephalic tincture. This the student was to swallow upon a fasting stomach, and for three days following eat nothing but bread and water. As the wafer digested the tincture mounted to the brain, bearing the proposition along with it.

- Jonathan Swift, Gulliver's Travels

Read:

1. Section 12. (Note that we are reading the text out-of-order here, so there a few words that we haven't defined yet. In particular, a "group" is a particular kind of binary structure, and the "dihedral group" $D_n$ is the symmetry group of the regular $n$-gon; we will clarify these points in the course of the next few lectures. You should still be able to skim this section and learn some interesting geometry; don't get hung up on details that aren't making sense yet.)

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

1. $\mathbb{Z}_n$.
2. $+_n$ (the operation of addition modulo $n$).
3. $\cdot_n$ (the operation of multiplication modulo $n$).
4. $\mathrm{Fun}(S,S)$.
5. $\circ$ (the operation of composition of functions).
6. $\mathrm{Iso}(\mathbb{R}^n)$.
7. $S(A)$ (the stabilizer or symmetry group of the geometric figure $A$).

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

1. Theorem concerning whether $+_n$ is well-defined.
2. Theorem concerning whether $\cdot_n$ is well-defined.
3. Theorem describing which elements of $(\mathbb{Z}_n,+_n)$ have inverses.
4. Theorem describing which elements of $(\mathbb{Z}_n,\cdot_n)$ have inverses (note that at this stage, the theorem is not very satisfying; we will make a better version a few months from now).
5. Theorem describing which elements of $(\mathrm{Fun}(S,S),\circ)$ have inverses.
6. Theorem describing which elements of $(\mathrm{Iso}(\mathbb{R}^n),\circ)$ have inverses.

Solve the following problems:

1. Section 12, problems 1, 4, 5, 6, and 7 (in problem 7, ignore the phrase "isomorphic to $\mathbb{Z}_4$" for now; i.e. just draw a figure with exactly four symmetries).
2. (Challenge) The chemical properties of a molecule turn out to be subtly influenced by the symmetry group of the molecule. The methane molecule is shaped like a regular tetrahedron; you can find a picture here. Try to describe all of the symmetries of the methane molecule. (Hint: in total there are twenty-four of them.)

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