Math 360, Fall 2020, Assignment 5

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. 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. Isometry (of $\mathbb{R}^n$).
5. $\mathrm{Iso}(\mathbb{R}^n)$.
6. $S(A)$ (the 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 placing $(\mathbb{Z}_n,+_n)$ in the "hierarchy of niceness" (i.e. as a semigroup, monoid, group, or abelian group).
4. Theorem describing placing $(\mathbb{Z}_n,\cdot_n)$ in the "hierarchy of niceness" (i.e. as a semigroup, monoid, group, or abelian group).

Solve the following problems:

1. Describe the group of units of each of the following monoids: $(\mathbb{Z}_4,\cdot_4)$, $(\mathbb{Z}_5,\cdot_5)$, $(\mathbb{Z}_6,\cdot_6)$, $(\mathbb{Z}_7,\cdot_7)$, and $(\mathbb{Z}_8,\cdot_8)$. Do you see any patterns?
2. 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).
3. (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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