Skip to main content
\( \newcommand{\lt}{ < } \newcommand{\gt}{ > } \newcommand{\amp}{ & } \)

Section8.4Exercises

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.)