Section8.4Exercises¶ permalink

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 $0\leq n < p$ you discovered using the interact in Subsection 8.2.1. This could be anything profounder than $\begin{equation*}a^0\equiv 1\text{ (mod }p)\text{ or }1^n\equiv 1\text{ (mod }p)\end{equation*}$ for all prime $p$ and for all $n$ , 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 $R^{-1}$ . How would you describe $R^3$ verbally? What is the order of $R$ ?

8

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

9

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

10

(Only if you have some experience with matrices.) Find two $2\times 2$ matrices $A$ and $B$ which have non-zero determinant such that $A\cdot B\neq B\cdot A$ . 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.)