Math 360, Fall 2014, Assignment 4

No doubt many people feel that the inclusion of mathematics among the arts is unwarranted. The strongest objection is that mathematics has no emotional import. Of course this argument discounts the feelings of dislike and revulsion that mathematics induces....

- Morris Kline, Mathematics in Western Culture

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

1. Subgroup (of a given group $G$).
2. Trivial subgroup.
3. Improper subgroup.
4. Cyclic subgroup (generated by a fixed element $a\in G$).
5. Cyclic group.
6. Generator (of a cyclic group).
7. Order (of a group).
8. Order (of an element of a group).
9. Quotient and remainder (resulting from integer division).
10. Greatest common divisor (of two integers).

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

1. Criterion for a subset to be a subgroup (Theorem 5.14).
2. Classification of cyclic groups (Theorem 6.10).
3. Theorem characterizing subgroups of cyclic groups (Theorem 6.6).
4. Classification of subgroups of $\mathbb{Z}$ (Theorem 6.7).
5. Classification of subgroups of $\mathbb{Z}_n$ (Theorem 6.14, though there are nicer ways to summarize this).

Solve the following problems:

1. Section 5, problems 2, 3, 11, 12, 13, 22, 23, and 27.
2. Section 6, problems 1, 3, 5, 13, 22, 23, and 24.

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