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MathGroup Archive 2006

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Re: too many special linear matrices->error in number

  • To: mathgroup at smc.vnet.net
  • Subject: [mg68716] Re: too many special linear matrices->error in number
  • From: Roger Bagula <rlbagula at sbcglobal.net>
  • Date: Thu, 17 Aug 2006 04:18:23 -0400 (EDT)
  • References: <ebuju2$6lf$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

SL[2,P]->
number of matrices in the group =If [n=2,6,Prime[n]*(Prime[n]^2-1)]
PSL[2,P]->
number of matrices in the group =If [n=2,6,Prime[n]*(Prime[n]^2-1)/2]
Half as many for PSL except for p=2...
Still the program is wrong for primes  5 and 7!
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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