Math 360, Fall 2018, Assignment 6

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

- Goethe

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

1. Theorem concerning the definition of modular addition.
2. Theorem concerning the number of solutions of the equation $a*x=b$.
3. Theorem concerning the number of solutions of the equation $x*a=b$.
4. Theorem concerning the number of occurrences of an element in a row or column of a group table.
5. Theorem concerning groups with two elements.
6. Theorem concerning groups with three elements.

Solve the following problems:

1. Section 4, problem 7.
2. (Multiplication modulo $n$) Define a binary operation $\cdot_n$ on $\mathbb{Z}_n$ by the formula $$[a]_n\cdot_n[b]_n = [ab]_n.$$ Prove that this operation is well-defined.
3. Make an operation table for the binary structure $(\mathbb{Z}_3,\cdot_3)$.
4. Show that the binary structure $(\mathbb{Z}_n,\cdot_n)$ is always a monoid but is never a group (except when $n=1$).
5. Make group tables for the groups of units $G(\mathbb{Z}_3,\cdot_3)$ and $G(\mathbb{Z}_4,\cdot_4)$.
6. Show that the groups of units you computed above are isomorphic to each other.
7. Perhaps the groups of units $G(\mathbb{Z}_n,\cdot_n)$ are all isomorphic to each other. Prove this, or give a counterexample.

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