Math 360, Fall 2016, Assignment 6

I tell them that if they will occupy themselves with the study of mathematics, they will find in it the best remedy against the lusts of the flesh.

- Thomas Mann, The Magic Mountain

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

1. Greatest common divisor (of two integers).

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

1. Theorem concerning subgroups of cyclic groups.
2. Theorem characterizing when $\left\langle a\right\rangle = \left\langle b\right\rangle$ in $\mathbb{Z}$.
3. Theorem characterizing when $\left\langle a\right\rangle = \left\langle b\right\rangle$ in $\mathbb{Z}_n$.
4. Classification of subgroups of $\mathbb{Z}$ (i.e. giving a non-redundant list of all subgroups).
5. Classification of subgroups of $\mathbb{Z}_n$.

Solve the following problems:

1. Section 6, problems 5, 23, 25, 33, and 34.
2. Section 7, problems 1, 3, and 5.

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