Number Theory: In Context and Interactive

1.

Write out the addition table for ${\mathbb{Z}}_{11}$ completely, by hand.

2.

Write out the multiplication table for ${\mathbb{Z}}_{11}$ completely, by hand.

3.

Find some conjecture/pattern to state about multiplication tables, based on any of the interacts in this chapter.

4.

Find some conjecture/pattern to state about values of $a^n$ (mod $p$), for $p$ prime and $0\le n<p$ you discovered using the interact in Subsection 8.2.1. This could be anything profounder than

for all prime $p$ and for all $n\text{,}$ but should at least be some pattern you tested for a number of values.

5.

Give an example of a non-closed binary operation.

6.

In Example 8.3.2, what is the order of the group element which is rotation by ninety degrees to the left? What is the order of rotation by 180 degrees?

7.

Consider a similar setup to Example 8.3.2, but with a regular hexagon. If $R$ is rotation of the hexagon by sixty degrees to the right, verbally describe $R^{-1}\text{.}$ How would you describe $R^3$ verbally? What is the order of $R\text{?}$

8.

Without using other resources, explain why Fact 8.3.7 is known as the “socks and shoes” property.

9.

Give an informal argument that $\mathbb{Q}$ is not cyclic.

10.

Give an example of a cyclic group which is not finite.

11.

(Only if you have some experience with matrices.) Find two $2\times 2$ matrices $A$ and $B$ which have non-zero determinant such that $A\cdot B\ne B\cdot A\text{.}$ Conclude that the group of $2\times 2$ matrices with non-zero determinant is not Abelian. (It is a group, because all such matrices have an inverse matrix.)