Chapter 3.1, Problem 34E

Let $G$ be the set of eight elements $G=\left\{1,i,j,k,-1,-i,-j,-k\right\}$ with identity element $1$ and noncommutative multiplication given by

${\left(-1\right)}^2=1,$

$i^2=j^2=k^2=-1,$

$ij=-ji=k$

$jk=-kj=i,$

$ki=-ik=j,$

$-x=\left(-1\right)x=x\left(-1\right)$ for all $x$ in $G$

(The circular order of multiplication is indicated by the diagram in Figure $3.8$.) Given that $G$ is a group of order $8$, write out the multiplication table for $G$. This group is known as the quaternion group. (Sec. $3.3,\#22a,32a,$

Sec. $3.4,\#2,$

Sec. $3.5,\#11,$

Sec. $4.2,\#8,$

Sec. $4.4,\#23,$

Sec. $4.5,\#40a,$

Sec. $4.6,\#3,11,16$)

Sec. $3.3\#22a,32a\ll$

22. Find the center $Z(G)$ for each of the following groups $G$.

a. $G=\left\{1,i,j,k,-1,-i,-j,-k\right\}$ in Exercise 34 of section 3.1.

32. Find the centralizer for each element $a$ in each of the following groups.

a. The quaternion group $G=\left\{1,i,j,k,-1,-i,-j,-k\right\}$ in Exercise 34 of section 3.1

Sec. $3.4,\#2\ll$

2. Let $G=\{\pm 1,\pm i,\pm j,\pm k\}$ be the quaternion group. List all cyclic subgroups of $G$.

Sec. $3.5,\#11\ll$

11. The following set of matrices

$\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$, $\left[\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right]$, $\left[\begin{array}{cc}0& 1\\ -1& 0\end{array}\right]$, $\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right]$

$\left[\begin{array}{cc}i& 0\\ 0& -i\end{array}\right]$, $\left[\begin{array}{cc}-i& 0\\ 0& i\end{array}\right]$, $\left[\begin{array}{cc}0& i\\ i& 0\end{array}\right]$, $\left[\begin{array}{cc}0& -i\\ -i& 0\end{array}\right]$ forms a group $H$ with respect to matrix multiplication. Find an isomorphism from $H$ to the quaternion group.

Sec. $4.2,\#8\ll$

8. Let be the quaternion group of units $U_{14}=\{\left[1\right],\left[3\right],\left[5\right],\left[9\right],\left[11\right],\left[13\right]\}\subseteq {\mathbb{Z}}_{14}$.

Sec. $4.4,\#23\ll$

23. Find all subgroups of the quaternion group.

Sec. $4.5,\#40a\ll$

40. Find the commutator subgroup of each of the following groups.

a. The quaternion group $\left\{\pm 1,\pm i,\pm j,\pm k\right\}$.

Sec. $4.6,\#3,11,16\ll$

3. The quaternion group $G=\left\{\pm 1,\pm i,\pm j,\pm k\right\}$

; $H=\{\pm 1\}$.

11. Find all homomorphic images of the quaternion group.

16. Repeat Exercise with the quaternion group $G=\{1,i,j,k,-1,-i,-j,-k\}$, the Klein four group $G^{\prime }=\{e,a,b,ab\}$, and $\varnothing :G\to G^{\prime }$ defined by

$\varnothing (1)=\varnothing (-1)=e$

$\varnothing (i)=\varnothing (-i)=a$

$\varnothing (j)=\varnothing (-j)=b$

$\varnothing (k)=\varnothing (-k)=ab$