Re: too many special linear matrices

*To*: mathgroup at smc.vnet.net*Subject*: [mg68711] Re: [mg68687] too many special linear matrices*From*: Andrzej Kozlowski <akoz at mimuw.edu.pl>*Date*: Thu, 17 Aug 2006 04:18:17 -0400 (EDT)*References*: <200608160736.DAA06175@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

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>