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Math Help - Abstract Algebra: Completing coset Multiplication Tables

  1. #1
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    Abstract Algebra: Completing coset Multiplication Tables

    onsider this group of six matrices:

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

    I = \left(\begin{array}{cc}1&0\\0&1\end{array}\right) A = \left(\begin{array}{cc}0&1\\1&0\end{array}\right) B = \left(\begin{array}{cc}0&1\\-1&-1\end{array}\right)

    C = \left(\begin{array}{cc}-1&-1\\0&1\end{array}\right) D = \left(\begin{array}{cc}-1&-1\\1&0\end{array}\right) K = \left(\begin{array}{cc}1&0\\-1&-1\end{array}\right)

    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 -> <R*, \bullet> by y f(x) = det(x) for any Matrix x /in G.

    Question:

    Complete the coset multiplication table for G/N (N being the Ker(f)) and Im(G) (a subgroup of <R*, MATH]\bullet[/tex]>

    G/N:
    _|_______
    \ \ \ \ \ \ \ \ |
    \ \ \ \ \ \ \ \ |
    \ \ \ \ \ \ \ \ |


    Image(G):
    _|_________
    \ \ \ \ \ \ \ \ |
    \ \ \ \ \ \ \ \ |
    \ \ \ \ \ \ \ \ |


    I got {I, B, D} for the Ker{f} (If that's correct.)

    Thanks. I appreciate any help in completing the tables.
    Last edited by Iceflash234; May 3rd 2010 at 11:58 AM.
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  2. #2
    Junior Member
    Joined
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    This is what I think the G/N chart would be:

    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.

    How does that look?
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