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] >> >

**References**:**too many special linear matrices***From:*Roger Bagula <rlbagula@sbcglobal.net>