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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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