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]

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