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Abstract alg - SL(2,Z3)

  1. Sep 29, 2004 #1
    i have a question about elements in SL(2,Z3), a,b,c,d are intergers and ad-bc=1 or Det [A]=1. i have to write all the matrices of this group and prove that I do have all of them.

    i know that only 3 elements exists in Z3 {[0],[1],[2]} with all others just being repeats. i.e. [-3]=0, [[4]=[1].

    i can write 24 elements with ad-bc=1,
    i.e. [[1,2],[2,2]] which is [1][2]-[2][2]=[2]-[4]=[-2]=[1]

    my problem is that i can't quite write WHY i have found all elements and they are no more, i was trying to appraoch i using contradiction but cant get started
  2. jcsd
  3. Sep 30, 2004 #2

    matt grime

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    There is a well known formula for the formula of order of finite chevalley groups and finite lie groups, of whic SL(2,Z_3) is one.

    The first row may be any of 8 non-zero row vectors, ie a and b can be any pair except 0,0. Now, for each pair, one of the entries must be non-zero, you may now insert any of the elements of Z3 in the slot beneath this non-zero one, and this determines what the remaining 4th entry must be. hence counting them there are 8*3=24 elements.
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