Math 360, Fall 2020, 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

Read:

1. Section 6.

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. 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 every cyclic group is abelian.
3. Prove that every cyclic group is countable (i.e. either finite or countably infinite).
4. Show that each of the following subgroups of $(\mathbb{Z},+)$ can be generated by a single non-negative integer: (a) $\left\langle 4, 6\right\rangle$, (b) $\left\langle 15, 35\right\rangle$, and (c) $\left\langle 12, 18, 27\right\rangle$.
5. (Challenge) Following the pattern of the three parts of the last problem, try to guess a general formula for a single non-negative generator for the subgroup $\left\langle k_1,k_2,\dots,k_m\right\rangle$ of $(\mathbb{Z},+)$.

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