Re: too many special linear matrices

• To: mathgroup at smc.vnet.net
• Subject: [mg68722] Re: [mg68687] too many special linear matrices
• From: "Carl K. Woll" <carlw at wolfram.com>
• Date: Thu, 17 Aug 2006 04:18:29 -0400 (EDT)
• References: <200608160736.DAA06175@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```Roger Bagula wrote:
> 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

According to your formula, you should be getting

6, 24, 120, 336

> instead of what I should get:
> 6,12,60,168

These are the number of different matrices of the *projective* special
linear group PSL(2,k).

The flaw in your Mathematica code is the use of Abs. Just remove the Abs
and you will get the appropriate matrices of SL[2,k].

Another, slower method to obtain these matrices is to use Reduce, e.g.:

p = 3;
r = Reduce[Mod[o l - m n, p] == 1 && 0 <= l < p &&
0 <= m < p && 0 <= n < p && 0 <= o < p,
{l, m, n, o}, Integers]

and the list of matrices:

s = {{l,m},{n,o}} /. {ToRules[r]}

Carl Woll
Wolfram Research

> 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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