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

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 $$ \langle i,j\ |\ R_1 , R_2 , R_3 , R_4 \rangle $$