Math 360, Fall 2016, 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. Substructure.
2. Subgroup.
3. Subgroup generated by a subset.
4. Finitely generated group.
5. Cyclic subgroup (generated by a given element).
6. Cyclic group.
7. Order (of an element).

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

1. Theorem concerning intersections and unions of subgroups.
2. Theorem concerning integer division.
3. Classification of cyclic groups.

Solve the following problems:

1. Section 5, problems 1, 2, 8, 9, 22, 23, and 27.
2. Section 6, problems 1, 3, 17, and 19.
3. Complete the proof of the classification of cyclic groups. (Specifically, recall that $G$ is a cyclic group generated by $a$, in which the powers of $a$ are not all distinct, and $n$ is the least positive integer for which $a^n=e$. We defined $\phi:\mathbb{Z}_n\rightarrow G$ by the formula $\phi([i]_{\equiv_n}) = a^i$, and we showed that this is indeed a well-defined function, whose output does not depend on the choice of the class representative $i$. To complete the proof, show that $\phi$ is injective, surjective, and preserves operations.)

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