too many special linear matrices

• To: mathgroup at smc.vnet.net
• Subject: [mg68687] too many special linear matrices
• From: Roger Bagula <rlbagula at sbcglobal.net>
• Date: Wed, 16 Aug 2006 03:36:18 -0400 (EDT)
• Sender: owner-wri-mathgroup at wolfram.com

```In an old group theory book they talk about special linear groups over
the modulo  of prime
Integers: SL[2,P]
The formula given is
number of matrices in the group  =If [n=2,6,Prime[n]*(Prime[n]^2-1)]
(Essentual Student Algebra, Volume 5 ,Groups, T.S. /Blyth and E.F.
Robertson,1986, Chapman and Hall,New York, page 14)
So I tried to  generate the elements of the group in Mathematica by a
search program for Determinant one
matrices.
I get:
6,24,124,348
instead of what I should get:
6,12,60,168
Since the famous Klein group SL[2,7] is one of these ,
it would help to have a set of elements for that group!

Mathematica code:
Clear[M, k, s]
M = {{l, m}, {n, o}};
k = 3
s =
Union[Delete[Union[Flatten[Table[Flatten[Table[Table[If[Mod[Abs[Det[M]],
k] - 1 == 0, M , {}], {l, 0,k - 1}], {m, 0, k - 1}], 1], {n, 0, k - 1},
{o, 0, k - 1}], 2]], 1]]
Dimensions[s]

```

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