Chapter 3.1, Problem 36E

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. $3.3,\#22b,32b\ll$

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

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

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

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

Sec. $4.1,\#22\ll$

22. Find an isomorphism from the octic group $D_4$ in Example $12$ of this section to the group $G^{\prime }=\left\{I_2,R,R^2,R^3,H,D,V,T\right\}$ in Exercise $36$ of Section $3.1$.

Sec. $4.6,\#14\ll$

14. Let $G=\left\{I_2,R,R^2,R^3,H,D,V,T\right\}$ be the multiplicative group of matrices in Exercise $36$ of section $3.1$, let $G\text{'}=\left\{1,-1\right\}$ under multiplication, and define $\varphi :G\to G\text{'}$ by

$\varphi \left(\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]\right)=ad-bc$.

Assume that $\varphi$ is an epimorphism, and find the elements of $K=$ ker $\varphi$.

Write out the distinct elements of $G/K$.

Let $\theta :G/K\to G\text{'}$ be the isomorphism described in the proof of Theorem $4.27$, and write out the values of $\theta$.