Math 360, Fall 2013, 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.
2. Improper subgroup.
3. Trivial subgroup.
4. Cyclic subgroup (generated by a particular element).
5. Cyclic group.
6. Generator (of a cyclic group).
7. Order (of a group).
8. Order (of an element of a group).
9. GCD (of two integers).
10. Relatively prime.

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

1. Theorem concerning integer division (Theorem 6.3 in the text).
2. Classification of cyclic groups (Theorem 6.10).
3. Classification of subgroups of \(\mathbb{Z}\) (Corollary 6.7).
4. Classification of subgroups of \(\mathbb{Z}_n\) (Theorem 6.14).

Do the following problems:

1. Section 5, problems 7, 9, 22, 23, and 42.
2. Section 6, problems 1, 3, 5, 9, 23, 27, and 30.

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

Solutions:

Definitions

1. Subgroup.

   Definition:

   Let \(H\) be a subset of a group \((G,*)\) \(H\) is also a subgroup of \(G\) if it is closed under \(*\), and it is a group under \(*\).

   Example:

   The integers under addition are a subgroup of the real numbers under addition.

   Non-Example:

   The positive integers are not a subgroup of the integers under addition, because the positive integers lack an identity and inverses.

2. Improper Subgroup.

   Definition:

   If we have a group \(G\), \(G\) is the improper subgroup of itself.

   Example:

   The integers with addition are the improper subgroup of the integers with addition.

   Non-Example:

   The even integers with addition are not the improper subgroup of the integers with additon.

3. Trivial Subgroup.

   Definition:

   If we have a group \(G\), then the trivial subgroup of \(G\) is the subgroup consisting of only the identity element.

   Example:

   The group \((0,+)\) is the trivial subgroup of the integers with addition.

   Non-Example:

   The even integers are not the trivial subgroup of the integers with addition.

4. Cyclic Subgroup.

   Definition:

   The cyclic subgroup of a group \(G\) generated by \(a\in G\) is the subgroup \(< a>\) of \(G\) such that \(< a > = \{a^n | n\ in \mathbb{Z}\}\)

   Example:

   Non-Example: