Finding the image and completing multiplication tables for G/N and Im(G)

[The_Iceflash]

Homework Statement

Consider this group of six matrices:

Let G = {I, A, B, C, D, K}, Matrix Multiplication>

$$I =\begin{bmatrix}1 & 0\\0 & 1\end{bmatrix}$$ $$A =\begin{bmatrix}0 & 1\\1 & 0\end{bmatrix}$$ $$B =\begin{bmatrix}0 & 1\\-1 & -1\end{bmatrix}$$

$$C =\begin{bmatrix}-1 & -1\\0 & 1\end{bmatrix}$$ $$D =\begin{bmatrix}-1 & -1\\1 & 0\end{bmatrix}$$ $$K =\begin{bmatrix}1 & 0\\-1 & -1\end{bmatrix}$$

Operation Table for this group:

_|I A B C D K
I |I A B C D K
A|A I C B K D
B|B K D A I C
C|C D K I A B
D|D C I K B A
K|K B A D C I

Define $$f:G\rightarrow$$ $$\left\langle\(R^{*}, \bullet\right\rangle$$ by f(x) = det(x) for any Matrix x $$\in$$ G.

Questions:

List all the elements in the image of G?

Complete coset multiplication tables for G/N (N being the Ker(f)) and Im(G) (a subgroup of <R*, $$\bullet$$>

Homework Equations

N/A

The Attempt at a Solution

I know the image of G is the range. I'm not exactly sure what to consider the range.

For the multiplication tables I know I'm to set it up like this but I'm not sure how to complete them. I appreciate any help. I do know that the Ker(f) is {I, B, D}.

G/N:
_|_______
|
|
|

Image(G):
_|_________
|
|
|

 

[Mark44]

Homework Statement

Consider this group of six matrices:

Let G = {I, A, B, C, D, K}, Matrix Multiplication>

$$I =\begin{bmatrix}1 & 0\\0 & 1\end{bmatrix}$$ $$A =\begin{bmatrix}0 & 1\\1 & 0\end{bmatrix}$$ $$B =\begin{bmatrix}0 & 1\\-1 & -1\end{bmatrix}$$

$$C =\begin{bmatrix}-1 & -1\\0 & 1\end{bmatrix}$$ $$D =\begin{bmatrix}-1 & -1\\1 & 0\end{bmatrix}$$ $$K =\begin{bmatrix}1 & 0\\-1 & -1\end{bmatrix}$$

Operation Table for this group:

_|I A B C D K
I |I A B C D K
A|A I C B K D
B|B K D A I C
C|C D K I A B
D|D C I K B A
K|K B A D C I

Define $$f:G\rightarrow$$ $$\left\langle\(R^{*}, \bullet\right\rangle$$ by f(x) = det(x) for any Matrix x $$\in$$ G.

Questions:

List all the elements in the image of G?

Complete coset multiplication tables for G/N (N being the Ker(f)) and Im(G) (a subgroup of <R*, $$\bullet$$>

Homework Equations

N/A

The Attempt at a Solution

I know the image of G is the range. I'm not exactly sure what to consider the range.

For the multiplication tables I know I'm to set it up like this but I'm not sure how to complete them. I appreciate any help. I do know that the Ker(f) is {I, B, D}.

G/N:
_|_______
|
|
|

Image(G):
_|_________
|
|
|

For the first part, go through all six matrices and calculate f(x) for each of them. For example, f(A) = -1.

 

[The_Iceflash]

So, the f(x)'s i receive are 1 and -1. Oh so that's what my image should be. I get that now. Thanks.

 

[The_Iceflash]

Any help on the tables from anyone would be greatly appreciated.

This is what I'm thinking for the G/N table:

__|N Na
N| N Na
Na| Na N

I found only 2 cosets and one is the Kernel and the other one I called Na due to a being one of the elements in it.