Re: too many special linear matrices

• To: mathgroup at smc.vnet.net
• Subject: [mg68724] Re: [mg68687] too many special linear matrices
• From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
• Date: Thu, 17 Aug 2006 04:18:31 -0400 (EDT)
• References: <200608160736.DAA06175@smc.vnet.net> <CE14B3F1-9562-4363-9D03-D37E82CF28FB@mimuw.edu.pl>
• Sender: owner-wri-mathgroup at wolfram.com

```The code below will produce the list of elements of SL[n,p] (as long
as n and p are not too large):

SL[n_, p_] := Module[{vars =
Table[Unique[
a], {n^2}], iters, mat}, iters = Map[{#, 0, p - 1} &, vars]; mat =
Partition[vars, n]; Reap[
Do[If[Det[mat,
Modulus -> p] == 1, Sow[mat], Continue[]], Evaluate[Sequence @@
iters]]][[2, 1]]]

For n=2 it gives the lengths:

Table[Length[SL[2,Prime[i]]],{i,1,7}]

{6,24,120,336,1320,2184,4896}

which agree with the formula I sent earlier:

In[16]:=
Table[f[2,Prime[i]],{i,1,7}]

Out[16]=
{6,24,120,336,1320,2184,4896}

Andrzej Kozlowski

n 16 Aug 2006, at 14:15, Andrzej Kozlowski wrote:

> The order of the special linear group with entries in F_q --the
> field with q elements (where q is a power of a prime) is well known
> to be:
>
> f[n_, q_] := (1/(q - 1))*Product[q^n - q^i, {i, 0, n - 1}]
>
> This gives:
>
>
> Table[f[n,2],{n,1,5}]
>
> {1,6,168,20160,9999360}
>
>
> Table[f[n,3],{n,1,5}]
>
> {1,24,5616,12130560,237783237120}
>
> The list of results you quote in your post seems to be a union of
> these two (corresponding to taking coefficients in the integers
> module 2 and modulo 3).   Bu how on earth did you get 60 using the
> formula you quote from Blyth and Robertson?
>
> In any case, this formula seems to be simply  f[2,p] (p any prime)
> which you have somehow managed to misstate.
>
> Simplify[f[2, p]]
>
>
> p*(p^2 - 1)
>
>
>
> Andrzej Kozlowski
>
>
>
>
> On 16 Aug 2006, at 09:36, 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
>> 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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