NTIC Exercises

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Exercises 8.4 Exercises

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1.

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

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2.

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

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3.

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

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

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5.

Give an example of a non-closed binary operation.

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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?

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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 $R\text{?}$

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8.

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

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9.

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

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10.

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

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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 $A\cdot B\neq B\cdot A\text{.}$ 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.)