Math 360, Fall 2017, 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 7 (you may omit the material on "Cayley Digraphs" if you wish).
2. Section 8.

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

1. $\mathrm{gcd}(a,b)$ (for integers $a,b$).
2. $\mathrm{lcm}(a,b)$ (for integers $a,b$).
3. Permutation (of a set $S$).
4. $S_n$.
5. Two-row notation (for elements of $S_n$).
6. Order (of a group).

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

1. Properties of $\mathrm{gcd}(a,b)$ (one part of this theorem justifies the phrase "common divisor" while the other gives the precise meaning of the adjective "greatest").
2. Properties of $\mathrm{lcm}(a,b)$ (one part of this theorem justifies the phrase "common multiple" and the other gives the precise meaning of the adjective "least").
3. Criterion for one subgroup of $\mathbb{Z}$ to be contained in another.
4. Criterion for two subgroups of $\mathbb{Z}$ to be equal.
5. Classification of subgroups of $\mathbb{Z}$ (i.e. giving a unique preferred generator for each subgroup).
6. Criterion for one subgroup of $\mathbb{Z}_n$ to be contained in another.
7. Criterion for two subgroups of $\mathbb{Z}_n$ to be equal.
8. Classification of subgroups of $\mathbb{Z}_n$ (i.e. giving a unique preferred generator for each subgroup).
9. Formula for the order of $S_n$.

Solve the following problems:

1. Section 6, problems 9, 13, 14, 15, 17, 19, 22, 23, 24, 25, 27, and 29.
2. Section 7, problems 1, 3, and 5.
3. Section 8, problems 1, 3, 5, 6, 7, 8, and 9.
4. For an integer $a$, what is $\mathrm{gcd}(a,0)$? Refer carefully to the definition before you decide. What about $\mathrm{lcm}(a,0)$?

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