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Math Help - Find |X| such that |X| is not 0, and is not divisible by a prime p in N

  1. #1
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    Find |X| such that |X| is not 0, and is not divisible by a prime p in N

    Hi, the problem I don't know is this one below... I have also solved a similar one below. Thanks a lot for any help you can give me! I am so lost... I don't have a clue how to do this

    Suppose that X is the set of all 2*2 matrices

    (a | b)
    (c | d)

    (sorry, I don't know how to do matrices)

    such that a, b, c, d belong to {0, 1, 2, ..., p-1} and p does not divide (ad-bc) and ad-bc is not equal to zero either, where p is a prime in the set of natural numbers N.

    Find |X|.


    Well, I have a similar problem (I needed to find the number of possible matrices) solved when a, b, c, d belong to Q (set of irrational numbers) and where ad-bc=0, but how do I get it for the case above (with very different conditions)?

    When they belong to Q, I got:

    Set Q as finite and countable, q=|Q|
    Let q=Q

    Consider to matrices:

    (0 | 0)
    (c | d), so we have q^2 matrices

    Also consider

    (a | b)
    (c | d), where (a | b) is not equal to (0 | 0)

    If a \= 0, then (c | d) = (c/a)*(a | b)
    If b \= 0, then (c | d) = (d/a)*(a | b)

    (c | d) = n*(a | b), where n belongs to Q.

    So we have q*(q^2-1) matrices.

    Adding them gives: q(q^2-1)+q^2=q^3+q^2-q.
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  2. #2
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    Quote Originally Posted by mbmstudent View Post
    Hi, the problem I don't know is this one below... I have also solved a similar one below. Thanks a lot for any help you can give me! I am so lost... I don't have a clue how to do this

    Suppose that X is the set of all 2*2 matrices

    (a | b)
    (c | d)

    (sorry, I don't know how to do matrices)

    such that a, b, c, d belong to {0, 1, 2, ..., p-1} and p does not divide (ad-bc) and ad-bc is not equal to zero either, where p is a prime in the set of natural numbers N.

    Find |X|.


    In short, you want |GL(2,\mathbb{F}_p)| , since any element in X is an automorphism of the vector

    space \left(\mathbb{F}_p\right)^2 over \mathbb{Z}_p .

    Well, just count how many ordered basis are there for the above vector space (it is (p^2-1)(p^2-p)=p(p^2-1)(p-1))

    Tonio



    Well, I have a similar problem (I needed to find the number of possible matrices) solved when a, b, c, d belong to Q (set of irrational numbers) and where ad-bc=0, but how do I get it for the case above (with very different conditions)?

    When they belong to Q, I got:

    Set Q as finite and countable, q=|Q|
    Let q=Q

    Consider to matrices:

    (0 | 0)
    (c | d), so we have q^2 matrices

    Also consider

    (a | b)
    (c | d), where (a | b) is not equal to (0 | 0)

    If a \= 0, then (c | d) = (c/a)*(a | b)
    If b \= 0, then (c | d) = (d/a)*(a | b)

    (c | d) = n*(a | b), where n belongs to Q.

    So we have q*(q^2-1) matrices.

    Adding them gives: q(q^2-1)+q^2=q^3+q^2-q.
    .
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