[GAP Forum] conjugate elements

[Stefan Kohl]

Dear Busra Guven,

> I want to find all elements of a group that makes two specific elements
> ( that is I know they are conjugate by an element) conjugate?
> Is there such an algorithm?
> I found the algorithm RepresentativeAction(g,x,y) which gives only one
> element that makes x and y conjugate. Can I find all such elements?

Given a finite group G and elements g and h of G, the following function
returns a list of all elements x of G such that g^x = h:

AllConjugators := function ( G, g, h )
  return AsList(RightCoset(Centralizer(G,g),RepresentativeAction(G,g,h)));
end;

For example you can do the following:

gap> G := SymmetricGroup(8);;
gap> l := AllConjugators(G,(1,2,3,4,5)(6,7,8),(1,2,3)(4,5,6,7,8));
[ (1,4,7,2,5,8,3,6), (1,8,3,5,7,2,4,6), (1,4,7)(2,5,8)(3,6),
  (1,7,2,8,3,4,5,6), (1,8,2,4,6,3,5,7), (1,4,7,3,6,2,5,8), (1,6)(2,7)(3,8),
  (1,7)(2,8)(3,4,5,6), (1,8)(2,4,6)(3,5,7), (1,5,4,8,3,7,2,6), (1,6,3,8,2,7),
  (1,7,3,4,5,6,2,8), (1,5,4,8,2,6,3,7), (1,6,2,7,3,8), (1,5,4,8)(2,6)(3,7) ]
gap> List(l,g->(1,2,3,4,5)(6,7,8)^g);
[ (1,2,3)(4,5,6,7,8), (1,2,3)(4,5,6,7,8), (1,2,3)(4,5,6,7,8),
  (1,2,3)(4,5,6,7,8), (1,2,3)(4,5,6,7,8), (1,2,3)(4,5,6,7,8),
  (1,2,3)(4,5,6,7,8), (1,2,3)(4,5,6,7,8), (1,2,3)(4,5,6,7,8),
  (1,2,3)(4,5,6,7,8), (1,2,3)(4,5,6,7,8), (1,2,3)(4,5,6,7,8),
  (1,2,3)(4,5,6,7,8), (1,2,3)(4,5,6,7,8), (1,2,3)(4,5,6,7,8) ]

Hope this helps,

    Stefan Kohl

-----------------------------------------------------------------------------
http://www.gap-system.org/DevelopersPages/StefanKohl/
-----------------------------------------------------------------------------