Chapter 4.6, Problem 15E

Repeat Exercise $14$

with $G=\{I_2,M_1,M_2,M_3,M_4,M_5\}$, the multiplicative group of matrices in Exercise $17$ of Section $3.3$.

Let $G=\{I_2,R,R^2,R^3,H,D,V,T\}$ be the multiplicative group of matrices in Exercise $36$ of Section $3.1$, let $G^{\prime }=\{1,-1\}$ under multiplication, and define $\varnothing :G\to G^{\prime }$

by

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

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

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

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