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The_Iceflash
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Homework Statement
Consider this group of six matrices:
Let G = {I, A, B, C, D, K}, Matrix Multiplication>
[tex]I =\begin{bmatrix}1 & 0\\0 & 1\end{bmatrix}[/tex] [tex]A =\begin{bmatrix}0 & 1\\1 & 0\end{bmatrix}[/tex] [tex]B =\begin{bmatrix}0 & 1\\-1 & -1\end{bmatrix}[/tex]
[tex]C =\begin{bmatrix}-1 & -1\\0 & 1\end{bmatrix}[/tex] [tex]D =\begin{bmatrix}-1 & -1\\1 & 0\end{bmatrix}[/tex] [tex]K =\begin{bmatrix}1 & 0\\-1 & -1\end{bmatrix}[/tex]
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 [tex] f:G\rightarrow[/tex] [tex]\left\langle\(R^{*}, \bullet\right\rangle[/tex] by f(x) = det(x) for any Matrix x [tex]\in[/tex] 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*, [tex]\bullet[/tex]>
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):
_|_________
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