Math 360, Fall 2017, 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 3.
2. Section 4.

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

1. Substructure (of a binary structure).
2. Hereditary property.
3. Isomorphism (from one binary structure to another).
4. Isomorphic (binary structures).
5. Structural property.
6. Inverse (of an element of a binary structure with identity $e$).
7. Semigroup.
8. Monoid.
9. Group.

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

1. Theorem concerning the definition of modular addition.
2. Theorem concerning the solutions (in some group $(G,\triangle)$) of the equations $a\triangle x=b$ and $x\triangle a = b$.
3. Corollary concerning the uniqueness of inverse elements.
4. Corollary concerning the rows and columns of a finite group table.
5. Classification of groups with two elements ("Every group with two elements is isomorphic to...").
6. Classification of groups with three elements.
7. Counterexample to the obvious conjecture concerning groups with four elements.

Solve the following problems:

1. Section 3, problems 2, 3, 4, 5, 6, 10, 17, 30, 31, and 32.
2. Section 4, problems 1, 2, 3, 4, 5, 6, 8, 10, 11, 12, 13, and 14.
3. Prove that the existence of an identity element is a structural property.
4. Prove that the existence of inverses is a structural property.
5. Prove that $(\mathbb{Z}_2,+_2)$ is not isomorphic to $(\mathbb{Z}_3,+_3)$. (Note that this is not simply a matter of trying and failing to find an isomorphism; you must show convincingly that no one will ever find an isomorphism.)
6. A group with only one element is called a trivial group. Explain why people sometimes refer to "the" trivial group, instead of "a" trivial group.

--------------------End of assignment--------------------