> < ^ Date: Fri, 06 Jul 2001 14:16:56 +0300 (IDT)
^ From: Eugene Plotkin <plotkin@macs.biu.ac.il >
> < ^ Subject: question

Dear Forum,

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)\$.
```

2.Find in \$Sz(q)\$ (at least for some \$q\$) elements
\$x,z\$ of the same order such that

1) \$zx\$ is an involution,

```2) for some \$k\$ one has

\$\$
z^kx^2z^{-k}x^{-2}x^kx^{-2}z^{-k}=z^{-1}x^{-2}.
\$\$
```

It seems to us that the most appropriate tool to obtain
some experimental data is GAP 4. Unfortunately,
we have no GAP experts in our department and around.

If anybody is interested or just has time and experience,
any help or advice would be gratefully apreciated.

```With best wishes,
Eugene Plotkin. Bar Ilan University, Israel
```

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