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Math Help - Example of a group action

  1. #1
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    Example of a group action

    I'm trying to generate examples for group action on sets in the lines of first 2 chapters of Dummit & Foote. And maybe I'm doing something wrong.

    I tried to define group action like this:

    Take all 2x2 matrices with entries from Z2 and remove the ones with det=1. We've got 10 entries for the set. Let's call the set A.
    The remaining (2^2-1)(2^2-2)=6 2x2 matrices (the ones with det=1) form a group - GL(2,2).

    We then define a group action that maps GL(2,2) x A -> A by using plain matrix multiplication. Multiplication must work, since detAB=detA x detB.
    So if You multiply a matrix from GL(2,2) by matrix with det=0, You get matrix with det=0. Moreover:

    | 1 0 |
    | 0 1 | a = a for any a in A and

    g2 (g1 a) = (g2 g1) a for any g1, g2 in GL(2,2) and any a in A (this is simply the associativity of matrix multiplication).


    So, if we fix particular g in G, we get function that maps A to A bijectively - a permutation.

    Then the problem:


    | 0 0 | | 0 0 | | 0 0 |
    | 1 0 | | 0 0 | = | 0 0 |

    and

    | 0 0 | | 0 0 | | 0 0 |
    | 1 0 | | 1 1 | = | 0 0 |

    I'm starting to think I'm not multiplying right, or maybe somehow detAB=detA*detB does not work for matrices with elements from Z2
    (have little experience with matrices with elements not from R).

    Aghm, help???
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  2. #2
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    Re: Example of a group action

    What is the problem? You are multiplying two elements of A together. You are not multiplying anything from G.
    Thanks from mrproper and topsquark
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  3. #3
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    Re: Example of a group action

    Oh, boy ....
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