[GAP Forum] List all elements in a Weyl group using GAP.

[muhammad shah]

Dear  Jianrong,
I think you want the elements of B in permutation 
form so that you can work with them easily. If so, then it can be done 
easily as following
 gap> F:=FreeGroup("a", "b", "c");
<free group on the generators [ a, b, c ]>
gap> B:=F/[ F.1^2, F.2^2, F.3^2, (F.1*F.2)^4, (F.2*F.3)^3, (F.3*F.1)^2 ];
<fp group on the generators [ a, b, c ]>
gap> Size(B);
48
gap> Elements(B);
[ <identity ...>, a*b*a*b*c*b*a*b, c, c*b, a*b*a*b, b*a*b*c*b*a*b*c, a*b*a*b*c*b*a*b*c, a*b*a*c*b*a*b*c, b*a*b*c,
  a*c, b, a*c*b*a*b, b*a*b*c*b*a*b, b*c, a*c*b*a, a*b*a*b*c*b, a*c*b*a*b*c, a*b*a*b*c*b*a, c*b*a*b*c, a, a*b*a*c*b*a,
  b*c*b*a*b*c, b*a, c*b*a*b, b*c*b, a*b*a, a*b*c*b*a*b*c, a*b*a*b*c, b*a*c*b*a*b, a*b*c*b*a*b, b*a*b*c*b*a,
  a*b*a*c*b*a*b, b*a*b*c*b, c*b*a, b*a*b, a*b*c*b, b*c*b*a, a*b, b*a*c*b*a, a*b*a*c*b, b*a*c*b*a*b*c, a*b*a*c, a*b*c,
  b*a*c, a*c*b, b*a*c*b, a*b*c*b*a, b*c*b*a*b ]

gap> iso:=IsomorphismPermGroup(B);
[ a, b, c ] -> [ (2,3)(4,6)(8,10)(9,11), (1,2)(3,5)(6,8)(7,9)(11,12), (2,4)(3,6)(5,7)(8,9)(10,11) ]
gap> C:=Image(iso);
Group([ (2,3)(4,6)(8,10)(9,11), (1,2)(3,5)(6,8)(7,9)(11,12), (2,4)(3,6)(5,7)(8,9)(10,11) ])
gap> Elements(C);
[ (), (2,3)(4,6)(8,10)(9,11), (2,4)(3,6)(5,7)(8,9)(10,11), (2,6)(3,4)(5,7)(8,11)(9,10), (1,2)(3,5)(6,8)(7,9)(11,12),
  (1,2,4)(3,8,7)(5,9,6)(10,12,11), (1,2,5,3)(4,8,10,6)(7,9,12,11), (1,2,8,12,11,6)(3,4,5,9,10,7),
  (1,3,5,2)(4,6,10,8)(7,11,12,9), (1,3)(2,5)(4,10)(7,11)(9,12), (1,3,10,12,9,4)(2,6,5,11,8,7),
  (1,3,6)(2,10,7)(4,5,11)(8,12,9), (1,4,2)(3,7,8)(5,6,9)(10,11,12), (1,4)(3,9)(5,8)(6,7)(10,12),
  (1,4,9,12,10,3)(2,7,8,11,5,6), (1,4,7,6)(2,9,11,3)(5,8,12,10), (1,5)(4,8)(6,10)(7,12),
  (1,5)(2,3)(4,10)(6,8)(7,12)(9,11), (1,5,12,7)(2,8,9,4)(3,10,11,6), (1,5,12,7)(2,10,9,6)(3,8,11,4),
  (1,6,11,12,8,2)(3,7,10,9,5,4), (1,6,7,4)(2,3,11,9)(5,10,12,8), (1,6,3)(2,7,10)(4,11,5)(8,9,12),
  (1,6)(2,11)(4,7)(5,10)(8,12), (1,7,12,5)(2,4,9,8)(3,6,11,10), (1,7,12,5)(2,6,9,10)(3,4,11,8),
  (1,7)(2,9)(3,11)(5,12), (1,7)(2,11)(3,9)(4,6)(5,12)(8,10), (1,8)(3,9)(4,5)(6,12)(7,10),
  (1,8,3,12,6,9)(2,5,10,11,7,4), (1,8,7,10)(2,9,11,3)(4,12,6,5), (1,8,11)(2,12,6)(3,5,10)(4,9,7),
  (1,9,6,12,3,8)(2,4,7,11,10,5), (1,9)(2,7)(3,12)(5,11)(6,8), (1,9,10)(2,8,5)(3,4,12)(6,7,11),
  (1,9,5,11)(2,12,3,7)(4,8,10,6), (1,10,7,8)(2,3,11,9)(4,5,6,12), (1,10,9)(2,5,8)(3,12,4)(6,11,7),
  (1,10)(2,11)(4,12)(5,6)(7,8), (1,10,2,12,4,11)(3,5,8,9,7,6), (1,11,8)(2,6,12)(3,10,5)(4,7,9),
  (1,11,5,9)(2,7,3,12)(4,6,10,8), (1,11,4,12,2,10)(3,6,7,9,8,5), (1,11)(2,12)(3,7)(4,10)(5,9),
  (1,12)(2,8)(3,10)(4,9)(6,11), (1,12)(2,9)(3,11)(4,8)(5,7)(6,10), (1,12)(2,10)(3,8)(4,11)(6,9),
  (1,12)(2,11)(3,9)(4,10)(5,7)(6,8) ]
gap> Size(C);
48

Regards,
Muhammad Shah

> Date: Sat, 10 Nov 2012 19:50:04 +0800
> From: lijr07 at gmail.com
> To: forum at gap-system.org
> Subject: [GAP Forum] List all elements in a Weyl group using GAP.
> 
> Dear members of the Forum,
> 
> I use the following codes to list all elements in B3 Weyl group.
> 
> Are there some other methods which can obtain the list of elements in Weyl
> group in GAP? Thank you very much.
> 
> Best wishes,
> Jianrong.
> 
> F:=FreeGroup("a", "b", "c");
> <free group on the generators [ a, b, c ]>
> gap> B:=F/[ F.1^2, F.2^2, F.3^2, (F.1*F.2)^4, (F.2*F.3)^3, (F.3*F.1)^2 ];
> <fp group on the generators [ a, b, c ]>
> gap> Size(B);
> 48
> gap> Elements(B);
> [ <identity ...>, a*b*a*b*c*b*a*b, c, c*b, a*b*a*b, b*a*b*c*b*a*b*c,
>   a*b*a*b*c*b*a*b*c, a*b*a*c*b*a*b*c, b*a*b*c, a*c, b, a*c*b*a*b,
>   b*a*b*c*b*a*b, b*c, a*c*b*a, a*b*a*b*c*b, a*c*b*a*b*c, a*b*a*b*c*b*a,
>   c*b*a*b*c, a, a*b*a*c*b*a, b*c*b*a*b*c, b*a, c*b*a*b, b*c*b, a*b*a,
>   a*b*c*b*a*b*c, a*b*a*b*c, b*a*c*b*a*b, a*b*c*b*a*b, b*a*b*c*b*a,
>   a*b*a*c*b*a*b, b*a*b*c*b, c*b*a, b*a*b, a*b*c*b, b*c*b*a, a*b, b*a*c*b*a,
>   a*b*a*c*b, b*a*c*b*a*b*c, a*b*a*c, a*b*c, b*a*c, a*c*b, b*a*c*b,
> a*b*c*b*a,
>   b*c*b*a*b ]
> gap>
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