abstract alg - SL(2,Z3)

[ossito_the-diracian]

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

[matt grime]

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.