Math 360, Fall 2015, 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

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

1. Group of units (of a monoid).
2. Subgroup.
3. Subgroup generated by a subset.
4. Finitely generated group.
5. Cyclic group.
6. Cyclic subgroup (generated by a given element).
7. Order (of an element).

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

1. Theorem concerning the inverse of a product.
2. Theorem concerning intersections and unions of subgroups.
3. Theorem concerning integer division.
4. Classification of cyclic groups.

Solve the following problems:

1. Section 5, problems 1, 2, 8, 9, 22, 23, 27, and 28.
2. Section 6, problems 1, 3, 13, 14, 15 and 17.

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