Math 360, Fall 2019, Assignment 7

Do not imagine that mathematics is hard and crabbed and repulsive to commmon sense. It is merely the etherealization of common sense.

- Lord Kelvin

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

1. Generator (of a cyclic group).

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

1. Theorem concerning integer division.
2. Well-ordering property of the positive integers.
3. Classification of cyclic groups.

Solve the following problems:

1. Section 6, problems 1, 3, 9, 10, 17, 19, 33, 34, 35, 36, and 37.
2. Prove that $GL_1(\mathbb{R})$ is not cyclic. (Hint: think about its cardinality.)

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