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Math Help - field

  1. #1
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    field

    I have a question that I don't know the answer to.

    Let F be a field. I showed that SL2(F) defined to be the matrix
    (a b
    c d) where a,b,c,d are in F and ad-bc=1 is a group.

    I already proved that.

    However, the question goes on to ask,
    If F has q elements, how many elements are in the group SL2(F)?
    I don't know the answer.
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  2. #2
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    Quote Originally Posted by PvtBillPilgrim View Post
    I have a question that I don't know the answer to.

    Let F be a field. I showed that SL2(F) defined to be the matrix
    (a b
    c d) where a,b,c,d are in F and ad-bc=1 is a group.

    I already proved that.

    However, the question goes on to ask,
    If F has q elements, how many elements are in the group SL2(F)?
    I don't know the answer.
    Think about it, what you actually want to know are the elements that are units. Our good friend rgep has given an exlanation here.
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  3. #3
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    So it would just be:
    ((q^n)-1)*((q^n)-(q^n-1)) where n=2 in this case?

    Thus, ((q^2)-1)*((q^2)-(q))
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  4. #4
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    Quote Originally Posted by PvtBillPilgrim View Post
    So it would just be:
    ((q^n)-1)*((q^n)-(q^n-1)) where n=2 in this case?

    Thus, ((q^2)-1)*((q^2)-(q))
    No!

    (q-1)(q^2-q)(q^3-q^2)....(q^n-q^{n-1})
    Or if you want to be cool

    \prod_{k=1}^n (q^{k}-q^{k-1})
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  5. #5
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    I don't know the dimension?
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