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Order of SL(2,Zp), p is a prime.

  1. Jan 6, 2012 #1
    1. The problem statement, all variables and given/known data

    Determine the index [G:H], where H is a subgroup of the group G and;

    [itex] G = GL(2,Z_p) [/itex]

    [itex] H = SL(2,Z_p) [/itex]

    p is a prime

    Where [itex]GL(2,Z_p)[/itex] is the general linear group of 2x2 invertible matrices with entries in [itex]Z_p[/itex], [itex]SL(2,Z_p)[/itex] is the general linear group of 2x2 invertible matrices with entries in [itex]Z_p[/itex] but with determinant 1.

    2. Relevant equations

    [itex]|GL(2,Z_p)| = (p^2 -1)(p^2 -p) [/itex]

    Lagrange's theorem

    [itex]|GL(2,Z_p)| = [G:H]|SL(2,Z_p)| [/itex]

    3. The attempt at a solution

    To be able to do this, I need to intuitively work out the number of distinct cosets, or work out the order of [itex]SL(2,Z_p)[/itex] in terms of p.

    I know how to derive the order of [itex]GL(n,Z_p)[/itex] (note: not necessarily a 2x2 matrix), but doing the same for the special linear group doesn't seem to be as straightforward.

    I've looked at it this way, the determinant of [itex]SL(2,Z_p)[/itex] must be one, so

    [itex] ac-bd = 1 [/itex] - the determinant

    Thus if we let

    [itex] ac = m [/itex]

    Then

    [itex] bd = m - 1 [/itex]

    [itex] 1 \leq m \leq p^2 [/itex]

    So, as [itex] a [/itex] or [itex] c [/itex] cannot be zero, [itex] a [/itex] and [itex] c [/itex] can have values ranging from [itex] 1 [/itex] to [itex] p-1 [/itex]. i.e. [itex] a [/itex] can have [itex] (p-1) [/itex] different values and so can [itex] c [/itex]. So the number of matrices in [itex]SL(2,Z_p)[/itex] is

    (Number of combinations of a) times (Number of combinations of c) times (Number of combinations of b) times (Number of combinations of d)

    =

    [itex] (p-1) [/itex] times [itex] (p-1) [/itex] times (Number of combinations of b) times (Number of combinations of d)

    =

    [itex] (p-1)^2 [/itex] times (Number of combinations of b) times (Number of combinations of d)

    How would I work out the number of possible combinations of [itex] b [/itex] and [itex] d [/itex] from this? Once I work that out, I should times [itex] (p-1)^2 [/itex] by those to get the order of [itex]SL(2,Z_p)[/itex].
     
    Last edited: Jan 6, 2012
  2. jcsd
  3. Jan 6, 2012 #2

    micromass

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    Hmmm, what if you apply the isomorphism theorem to the map

    [tex]GL(2,\mathbb{Z}_p)\rightarrow \mathbb{Z}_p:A\rightarrow det(A)[/tex]

    ??
     
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