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# Abstract Algebra (3th ed.)/Chapter1/Exercises 1.5

## 11.03.2020

### 1)

In $Q_8$ we have the following elements $$\{ 1,-1,i,-i,j,-j,k,-k \}$$ Here $|1| = 1$. We have that $|-1| = 2$. We have that $|i| = |j| = |k| = 4$. Now $$(-i)^2 = -1 \cdot i \cdot -1 \cdot i = i \cdot -1 \cdot -1 \cdot i = i^2 = -1$$ So $|-i| = |-j| = |-k| = 4$. In total

 1 -1 i -i j -j k -k 1 2 4 4 4 4 4 4

### 2)

For the multiplication table we get

 x 1 -1 i -i j -j k -k 1 1 -1 i -i j -j k -k -1 -1 1 -i i -j j -k k i i -i -1 1 k -k -j j -i -i i 1 -1 -k k j -j j j -j -k k -1 1 i -i -j -j j k -k 1 -1 -i i k k -k j -j -i i -1 1 -k -k k -j j i -i 1 -1

### 3)

The group can be generated with two of the three letters. For example $i,j$. Here we need four of the relations stated in the book, that is

• $R_1 = (-1) \cdot (-1) = 1$
• $R_2 = (-1) \cdot a = a \cdot (-1) = -a$
• $R_3 = j \cdot i = -k$
• $R_4 = i \cdot i = -1$

In total we get $$\langle i,j\ |\ R_1 , R_2 , R_3 , R_4 \rangle$$

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