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

  1. May 3, 2010 #1
    1. The problem statement, all variables and given/known data
    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]>

    2. Relevant equations
    N/A

    3. 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):
    _|_________
    |
    |
    |
     
    Last edited: May 3, 2010
  2. jcsd
  3. May 3, 2010 #2

    Mark44

    Staff: Mentor

    For the first part, go through all six matrices and calculate f(x) for each of them. For example, f(A) = -1.
     
  4. May 3, 2010 #3
    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.
     
  5. May 4, 2010 #4
    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.
     
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