Dear Forum,

On Fri, Jul 06, 2001 at 02:16:56PM +0300, Plotkin Eugene wrote:
> We have a group theoretic question which requires some
> GAP experiments. Unfortunately, nobody around is
> experienced enough to get results. The problem is as follows.
> 
> 
> Let $x$, $y$ be two indeterminants. Consider the
> words  
>           $$u_1(x,y)=x^{-1}yxy^{-1)x$$
> and
>  $$u_2(x,y)=[xu_1x^{-1},yu_1y^{-1}]$$
> 
> where $[a,b]=aba^{-1}b^{-1}$.
> 
> By some reasons we are looking for the non-trivial 
> (i.e., $x$, $y$ differ from 1) solutions of the equation
> 
>                $$u_1=u_2$$
> 
> in the Suzuki groups $Sz(q)$, ($q=2^p$ with $p$ odd).
> 
> Keeping in mind this aim,  we would like to know:
> 
> 1. All solutions of $u_1=u_2$ for the small Suzuki groups
> $Sz(8)$, $Sz(32)$.
> 
at least for Sz(8), this is easy, as you can list all the
elements of the group:
gap> g:=SuzukiGroup(8); 
gap> c:=ConjugacyClasses(g);  
gap> r:=List(c,yy->Representative(yy));
gap> r:=Filtered(r,yy->yy<>id);
gap> elts:=Elements(g);;
gap> good:=[];;for x in r do           
	> for y in elts do
	> u1:=x^-1*y*x*y^-1*x;
> u2:=Comm(y*u1*y^-1,x*u1*x^-1);
> if u1=u2 then Add(good,[x,y]); fi;
> od;
> od;
gap> Length(good);                     
156

then e.g.
gap> good[11];                         
[ [ [ Z(2^3)^3, Z(2^3), Z(2)^0, Z(2^3)^5 ], 
      [ Z(2^3), Z(2)^0, Z(2^3)^6, Z(2^3)^3 ], 
      [ Z(2)^0, Z(2^3)^6, 0*Z(2), Z(2^3)^5 ], 
      [ Z(2^3)^5, Z(2^3)^3, Z(2^3)^5, Z(2^3)^6 ] ], 
  [ [ Z(2)^0, Z(2^3)^5, Z(2^3)^4, Z(2^3)^2 ], 
      [ Z(2^3)^3, Z(2^3)^3, Z(2^3)^2, Z(2^3)^5 ], 
      [ Z(2^3), Z(2^3), 0*Z(2), Z(2^3)^2 ], 
      [ Z(2^3)^4, Z(2^3)^2, Z(2)^0, Z(2^3)^4 ] ] ]
(note that in this way you can get in principle more than 1
representative of the orbits of g in the corresponding
action on ordered pairs of elements by conjugation)

Hope this helps,
Dmitrii

PS. Let me know if you need more details on this.