Math 360, Fall 2020, Assignment 8

Mathematical proofs, like diamonds, are hard as well as clear, and will be touched with nothing but strict reasoning.

- John Locke, Second Reply to the Bishop of Worcester

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

1. $\mathrm{gcd}(a,b)$.
2. $\mathrm{lcm}(a,b)$.

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

1. Classification of subgroups of cyclic groups ("Every subgroup of a cyclic group is...").
2. Containment criterion for subgroups of $\mathbb{Z}$.
3. Equality criterion for subgroups of $\mathbb{Z}$.
4. Classification of subgroups of $\mathbb{Z}$ ("Every subgroup of $\mathbb{Z}$ has a unique...").
5. Theorem concerning the properties of $\mathrm{gcd}(a,b)$.
6. Theorem concerning the properties of $\mathrm{lcm}(a,b)$.
7. Containment criterion for subgroups of $\mathbb{Z}_n$.
8. Theorem relating $\left\langle[a]\right\rangle$ to $\left\langle[\mathrm{gcd}(a,n)]\right\rangle$ (in $\mathbb{Z}_n$).
9. Classification of subgroups of $\mathbb{Z}_n$ ("Every subgroup of $\mathbb{Z}_n$ is generated by a unique...").

Solve the following problems:

1. Section 6, problems 5, 7, 22, 23, 24, and 25 (see page 51 and Example 6.17 for the meaning of the phrase subgroup diagram).
2. In some of the problems above, you drew subgroup diagrams for $\mathbb{Z}_n$ for various values of $n$. Now try to draw the subgroup diagram for $\mathbb{Z}$.
3. Prove that, in $\mathbb{Z}_n$, one always has $\left\langle\left[\mathrm{gcd}(a,n)\right]\right\rangle\subseteq\left\langle\left[a\right]\right\rangle$. (Hint: this is a straightforward application of the containment criterion.)
4. Prove that, in $\mathbb{Z}_n$, one always has $\left\langle\left[a\right]\right\rangle\subseteq\left\langle\left[\mathrm{gcd}(a,n)\right]\right\rangle$. (Hint: using the theorem concerning the properties of gcds, show that $\mathrm{gcd}(\mathrm{gcd}(a,n),n)=\mathrm{gcd}(a,n).$)

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