> < ^ Date: Sat, 07 Jul 2001 01:02:23 +0000
> < ^ From: Dmitrii Pasechnik <d.pasechnik@twi.tudelft.nl >
> < ^ Subject: Re: question

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.

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