Chapter 3.3, Problem 22E

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.

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

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

d. $G=GL(2,R)$, the general linear group of order $2$

over $R$.

Exercise 34 of section 3.1.

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.

Exercise 36 of section 3.1

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 $G$.

Exercise 35 of section 3.1.

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.

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

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

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

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

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

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 $G$.