Chapter 3.3, Problem 32E

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

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

$G=\left\{I_2,R,R^2,R^3,H,D,V,T\right\}$ in Exercise 36 of section 3.1 (Sec. 3.1, #36).

$G=\left\{I_3,P_1,P_2,P_3,P_4,P_5\right\}$ in Exercise 35 of section 3.1 (Sec. 3.1, #35).

Sec. $3.1,\#34\gg$

34. 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. $3.1,\#36\gg$

Consider the matrices

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

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

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

$D=\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]$

$T=\left[\begin{array}{cc}0& -1\\ -1& 0\end{array}\right]$ in $GL(2,ℝ)$, and let $G=\left\{I_2,R,R^2,R^3,H,D,V,T\right\}$. Given that $G$ is a group of order 8 with respect to multiplication, write out a multiplication table for $G$. (Sec. $3.3,\#22b,32b,$

Sec. $4.1,\#22,$

Sec. $4.6,\#14$)

Sec. $3.1,\#35\gg$

35. A permutation matrix is a matrix that can be obtained from an identity matrix $I_n$ by interchanging the rows one or more times (that is, by permuting the rows). For $n=3$ the permutation matrices are $I_3$ and the five matrices.

(Sec. $3.3,\#22c,32c$, Sec. $3.4,\#5$, Sec. $4.2,\#6$)

$P_1=\left[\begin{array}{ccc}1& 0& 0\\ 0& 0& 1\\ 0& 1& 0\end{array}\right]$

$P_2=\left[\begin{array}{ccc}0& 1& 0\\ 1& 0& 0\\ 0& 0& 1\end{array}\right]$

$P_3=\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ 1& 0& 0\end{array}\right]$

$P_4=\left[\begin{array}{ccc}0& 0& 1\\ 0& 1& 0\\ 1& 0& 0\end{array}\right]$

$P_5=\left[\begin{array}{ccc}0& 0& 1\\ 1& 0& 0\\ 0& 1& 0\end{array}\right]$

Given that $G=\left\{I_3,P_1,P_2,P_3,P_4,P_5\right\}$ is a group of order $6$ with respect to matrix multiplication, write out a multiplication table for $G$.