Math 360, Fall 2018, Assignment 9

The moving power of mathematical invention is not reasoning but the imagination.

- Augustus de Morgan

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

1. $\mathrm{gcd}(a,b)$ (where $a,b\in\mathbb{Z}$).
2. $\mathrm{lcm}(a,b)$ (where $a,b\in\mathbb{Z}$).
3. Divisibility relation (on $\mathbb{Z}$).

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

1. Theorem concerning subgroups of cyclic groups.
2. Theorem relating $\mathrm{gcd}(a,b)$ to common divisors of $a$ and $b$.
3. Theorem relating $\mathrm{lcm}(a,b)$ to common multiples of $a$ and $b$.
4. Containment criterion for subgroups of $\mathbb{Z}$.
5. Classification of subgroups of $\mathbb{Z}$ (i.e. giving a unique "preferred" generator for each subgroup).
6. Containment criterion for subgroups of $\mathbb{Z}_n$.
7. Classification of subgroups of $\mathbb{Z}_n$ (i.e. giving a unique "preferred" generator for each subgroup).

Solve the following problems:

1. Section 6, problems 22, 23, 24, 25, 27, and 29 (for problems 25-29, the order of a subgroup is the number of elements of that subgroup).
2. Suppose $p$ and $q$ are distinct prime numbers. Draw the subgroup diagram for $\mathbb{Z}_{pq}$.

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