%%% Date: Tue, 7 Nov 2000 14:59:44 +0200 (IST)
%%% From: Boris Kunyavski <kunyav@macs.biu.ac.il>
%%% X-Sender: kunyav@sunlight
%%% To: Ulf Rehmann <rehmann@Mathematik.Uni-Bielefeld.DE>
%%% Subject: preprint
%%% In-Reply-To: <200010312334.AAA20608@jcn09.Mathematik.Uni-Bielefeld.DE>
%%% X-Organization: Math & CS department
%%% Content-Type: MULTIPART/MIXED; BOUNDARY="-559023410-851401618-973601906=:954"
%%% Content-ID: <Pine.GSO.4.10_heb2.08.10011071458500.1948@sunlight>
%%% 
%%% Content-Type: TEXT/PLAIN; CHARSET=US-ASCII
%%% Content-ID: <Pine.GSO.4.10_heb2.08.10011071458501.1948@sunlight>
%%% 
%%% Dear Ulf, 
%%% 
%%%    You will find attached to this message: 
%%% 
%%% 1) a LaTeX file of our recent preprint we were talking 
%%% about this August; 
%%% 
%%% 2) an abstract; 
%%% 
%%% 3) a PostScript file. 
%%% 
%%%    We would like to put this preprint on LAG preprint 
%%% server and also to submit it to SFB343 preprint series. 
%%% 
%%%    Many thanks in advance, 
%%% 
%%%     B.Kunyavskii, E.Plotkin. 
%%%  
%%% 
%%% 
%%% Content-Type: APPLICATION/X-TEX; NAME="sfb.tex"
%%% Content-Transfer-Encoding: BASE64
%%% Content-ID: <Pine.GSO.4.10_heb2.08.10011071458260.954@sunlight>
%%% Content-Description: 
%%% Content-Disposition: ATTACHMENT; FILENAME="sfb.tex"
%%% 
%%% 
%%% This message contains data in an unrecognized format, application/x-tex,
%%% which is being decoded and written to the file named "/tmp/sfb.tex".
%%% If you do not want this data, you probably should delete that file.
%%% Wrote file /tmp/sfb.tex
%%% Content-Type: TEXT/PLAIN; CHARSET=US-ASCII; NAME=abstract
%%% Content-Transfer-Encoding: BASE64
%%% Content-ID: <Pine.GSO.4.10_heb2.08.10011071458261.954@sunlight>
%%% Content-Description: 
%%% Content-Disposition: ATTACHMENT; FILENAME=abstract
%%% 
%%% Authors: F. Grunewald, B. Kunyavskii, D. Nikolova, and E. Plotkin 
%%% 
%%% Title: Two-variable identities in groups and Lie algebras 
%%% 
%%% Email: plotkin@macs.biu.ac.il
%%% 
%%% 1991 Mathematics Subject Classification: 17B01, 17B30, 20F16, 20G15 
%%% 
%%% Keywords and Phrases: identity, solvable group, solvable Lie algebra 
%%% 
%%% Abstract Text: 
%%% 
%%% We study two-variable Engel-like relations and identities 
%%% characterizing finite dimensional solvable Lie algebras and, 
%%% conjecturally, finite solvable groups, and introduce some 
%%% invariants of finite groups associated to such relations. 
%%% 
%%% 
%%% Content-Type: APPLICATION/postscript; name="sfb.ps"
%%% Content-Transfer-Encoding: BASE64
%%% Content-ID: <Pine.GSO.4.10_heb2.08.10011071459440.1948@sunlight>
%%% Content-Description: 
%%% Content-Disposition: attachment; filename="sfb.ps"
%%% 
%%% 
%%% This message contains data in an unrecognized format, application/postscript,
%%% which is being decoded and written to the file named "/tmp/sfb.ps".
%%% If you do not want this data, you probably should delete that file.
%%% Wrote file /tmp/sfb.ps
%%% 
%%% %This is LaTeX
%%% %29.09.2000
%%% 
\documentstyle[12pt,a4,amssymb,xypic]{article}
\pagestyle{plain}

\oddsidemargin 0 cm
\evensidemargin 2 cm
\textwidth 17 cm
%\topmargin 0 cm
%\textheight 20.5 cm

\newtheorem{df}{Definition}[section]
\newtheorem{prob}[df]{Problem}
\newtheorem{prop}[df]{Proposition}
\newtheorem{thm}[df]{Theorem}
\newtheorem{lem}[df]{Lemma}
\newtheorem{conj}[df]{Conjecture}



 \def\today{\ifcase\month\or
  January\or February\or March\or April\or May\or June\or
  July\or August\or September\or October\or November\or December\fi
  \space\number\day, \number\year}

\begin{document}


\title{
{Two-variable identities in groups and Lie algebras}
\thanks{This research was supported by the Israel Science Foundation founded 
by the Israel Academy of Sciences and Humanities --- Center of Excellence 
Program.}
}

\author{
{\sc Fritz Grunewald} \\
\small Math. Institut, Heinrich Heine Universit\"at, \\
\small 40225 D\"usseldorf, GERMANY \\
\small e-mail: grunewald@math.uni-duesseldorf.de \\
{}\\
{\sc Boris  Kunyavski\u\i 
\thanks{Partially supported by the Ministry of Absorption (Israel) and the Minerva 
Foundation through the Emmy Noether Research Institute of Mathematics.}}   \\
\small Dept of Math. and Computer Science, Bar-Ilan University,    \\
\small  52900 Ramat Gan, ISRAEL  \\
\small  e-mail: kunyav@macs.biu.ac.il \\  
{} \\
{\sc Daniela Nikolova} \\
\small Insitute of Math., Bulgarian Academy of Sciences, \\
\small Sofia, BULGARIA \\
\small e-mail: daniela@moi.math.bas.bg \\  
{} \\
%{\sc and} \\
%{} \\
{\sc Eugene Plotkin$^\dagger$ } \\
\small Dept of Math. and Computer Science, Bar-Ilan University,    \\
\small  52900 Ramat Gan, ISRAEL  \\
\small  e-mail: plotkin@macs.biu.ac.il 
}

%\footnotetext{Partially supported by the Ministry of Absorption (Israel) and the Minerva 
%Foundation through the Emmy Noether Insitute of Mathematics.}

\date{}


\maketitle

\thispagestyle{empty}
 
\begin{abstract} 
We study two-variable Engel-like relations and identities 
characterizing finite dimensional solvable Lie algebras 
and, conjecturally, finite solvable groups, and introduce some  
invariants of finite groups associated to such relations.  
\end{abstract}

\newpage




%Bbb
\def\Gm{{{\Bbb G}}_{\rm{m}}}
\def\GmF{{{\Bbb G}}_{{{\rm{m}},F}}}
\def\GmL{{{\Bbb G}}_{{{\rm{m}},L}}}
\def\Gmr{{{\Bbb G}}_{{\rm{m}}}^r}
\def\Gmk{{{\Bbb G}}_{{{\rm{m}},k}}}
\def\Z{{\Bbb Z}}
\def\R{{\Bbb R}}




%Greek
\def\alp{{\alpha}}  \def\bet{{\beta}} \def\gam{{\gamma}}
 \def\del{{\delta}}
\def\eps{{\varepsilon}}
\def\kap{{\kappa}}                   \def\Chi{\text{X}}
\def\lam{{\lambda}}
 \def\sig{{\sigma}}  \def\vphi{{\varphi}} \def\om{{\omega}}
\def\Gam{{\Gamma}}  \def\Del{{\Delta}}  \def\Sig{{\Sigma}}
\def\ups{{\upsilon}}

%Broman
\def\Gal{{\rm{Gal}}}          
\def\Aut{{\rm{Aut\,}}}
\def\A{{\rm{A}}}
\def\P{{\rm{P}}}
\def\Q{{\rm{Q}}}
\def\W{{\rm{W}}}
\def\ker{{\rm{ker\,}}}
\def\Hom{{\rm{Hom}}}
\def\Pic{{\rm{Pic\,}}}
\def\Sym{{\rm{Sym}}} 
\def\nor{{\rm{N}}}
%\def\rg{{\rm{rg\,}}}
\def\rk{{\rm{rk\,}}}
\def\GL{{\rm{GL}}}
\def\SL{{\rm{SL}}}
\def\PSL{{\rm{PSL}}}
\def\Sz{{\rm{Sz}}}
\def\sign{{\rm sign}}
\def\im{{\rm{im\,}}}
\def\Ind{{\rm{Ind}}}
\def\Id{{\rm{Id}}}
\def\id{{\rm{id}}}
\def\out{_{\rm out}}
\def\sep{_{\rm sep}}
\def\et{_{{\rm{{\acute e}t}}}}
\def\char{{\rm char}}
\def\pr{{\rm{pr\,}}}
\def\rad{{\rm rad}}


\def\min{^{-1}}
\def\ss{^{{\rm{ss}}}}
\def\red{^{{\rm{red}}}}
\def\un{^{{\rm{u}}}}

%gothic
\def\gg{{\frak g}}
\def\hh{{\frak h}}


%\hats
\def\Th{{\hat T}}

%\bars
\def\Xb{{\overline X}}
\def\Vb{{\overline V}}
\def\Fb{{\overline F}}
\def\Lb{{\overline L}}

\def\kb{{\bar k}}
\def\xb{{\bar x}}
\def\yb{{\bar y}}






\def\into{{\hookrightarrow}}


%Group schemes
\def\bG{{\cal G}}
\def\bE{{\cal E}}
\def\bSL{{\cal S}{\cal L}}
\def\bPSL{{\cal P}{\cal S}{\cal L}}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Motivation} \label{sec:motiv}

Our primary interest in problems discussed in the present paper 
came from a paper by Segev \cite{Se} where the 
Margulis--Platonov conjecture had been related to some properties 
of the commuting graph of a finite simple group. (Given a finite 
group $F$, its commuting graph $\Gam (F)$ has vertices indexed by  
the elements of $F$ different from 1;  
$x,y\in F$ are joined by an edge if and only if they commute.) 
In a more recent paper \cite{SS} the expected properties of this graph 
have been proved. 

In its simplest form, the Margulis--Platonov conjecture asserts that 
if $G$ is any (absolutely almost) simple linear algebraic group 
defined over a number field $k$ then the group of rational points 
$G(k)$ contains no non-central normal subgroups if and only if the 
same is true for all groups $G(k_v)$ where $v$ runs over all places of 
$k$ and $k_v$ stands for the completion of $k$ at $v$ (see 
\cite[9.1]{PlR}). The most difficult 
case of this conjecture is that of anisotropic groups of type $A_n$. 
In the case of the inner forms, Potapchik and Rapinchuk \cite{PoR} 
reduced the 
conjecture to a purely algebraic statement that the multiplicative 
group of any finite dimensional division $k$-algebra has no 
non-abelian finite simple quotients. For this purpose, Segev proved 
\cite{Se} that if $D$ is a finite dimensional division algebra  
over an arbitrary field and $F$ is a finite simple non-abelian 
group whose commuting graph $\Gam (F)$ is either balanced (see 
\cite{Se} for the definition), or is of diameter greater than 4, 
then $F$ cannot be a quotient of $D^*$. In \cite{SS}, it is 
proved that the commuting graph of any finite non-abelian simple group 
is either balanced, or of diameter greater than 4. This completes 
the proof. 
 
All above shows that $\Gam (F)$ is a powerful invariant of $F$. 
After Segev's lecture on this topic, B.~Plotkin suggested two 
natural generalizations of the commuting graph of a group. Namely, 
it is natural to consider graphs of nilpotency and solvability 
of an arbitrary group $G$. To define them, we need to formulate 
conditions of nilpotency and solvability as two-variable relations 
between the elements of $G$. This is done in Section \ref{sec:inv}. 
In Section \ref{sec:Lie} we consider Lie-algebraic analogues of 
these conditions. In Section \ref{sec:lin} we focus on the case 
of linear algebraic groups and groups of their rational points. 
In Section \ref{sec:conj} we return to the case of finite groups 
which was the main motivation for this paper. 

\bigskip

\noindent
{\it Notation}. If $G$ is a group, $x,y\in G$, let  
$[x,y]=x\min y\min xy$. If $L$ is a Lie algebra, $x,y\in L$, 
we use the same symbol $[x,y]$ for the Lie product. 

\bigskip

{\it Acknowledgements}. This work was done while D.~Nikolova 
was visiting Bar-Ilan University and Ben-Gurion University 
of the Negev, Israel, and the National University of Ireland, 
Galway. The paper was finished while B.~Kunyavski\u\i \ 
and E.~Plotkin were visiting SFB 343 ``Diskrete Strukturen in der 
Mathematik'' (University of Bielefeld). The authors gratefully 
appreciate hospitality and support of these institutions. 
We thank M.~Borovoi, R.~Farnsteiner, A.~Premet, and N.~Vavilov for 
very useful discussions and correspondence. Our special 
thanks go to B.~Plotkin whose ideas and encouragement were 
indispensable, and to A.~Feldman and R.~Shklyar for their help in computer 
experiments.    


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{New invariants of finite groups} 
\label{sec:inv}

Our primary goal is to introduce some new invariants of a  
group $G$ associated to two-variable relations between 
elements of $G$. 

\begin{df} Let $G$ be an arbitrary group, and let 
$\rho$ be a binary relation on $G$. 
We define a (directed) graph $\Gam ^+ =\Gam ^+_{\rho } (G)$ 
($\rho$-graph of $G$) as follows: the vertices of $\Gam ^+$ 
are indexed by the elements of $G$ different from 1, 
and vertices $g,h$ form an edge directed from $g$ to $h$ 
if and only if $g\neq h$ and $g\rho h$. We denote by $\Gam$ 
the non-directed graph obtained from $\Gam ^+$ by 
forgetting orientation and deleting multiple edges. 
\end{df}

\noindent
{\it Remark}. If $\rho$ is symmetric, we only consider 
the non-directed graph $\Gam _{\rho }(G)$.  

\noindent
{\it Example}. If $\rho$ is the commuting relation 
(i.e. $g\rho h$ if and only if $gh=hg$), we get 
the commuting graph of $G$ studied in \cite{Se}. 

\bigskip
 
First we consider ``nilpotency relations''. Denote Engel 
words by $v_1=v_1(x,y)=[x,y]$, $v_2=[v_1,y], \dots ,
v_n=[v_{n-1},y],\dots$. 

\begin{df} \label{def:rel-nil}
Let $G$ be an arbitrary group. We say that 
elements $g,h\in G$ are in $n$-Engel relation $\nu _n$ 
if $v_n(g,h)=1$. Elements $g,h$ are said to be in 
nilpotency relation $\nu$ if they are in $n$-Engel 
relation for some $n$. 
\end{df}

Recall that a finite group $G$ is nilpotent if and only 
if the identity $v_n(x,y)\equiv 1$ holds in $G$ for some $n$, 
or, in terms of the above definition, if and only if 
$g\nu h$ for all $g,h\in G$. Note that $v_k\equiv 1$ implies 
$v_l\equiv 1$ for all $l>k$.  

\begin{df} \label{def:graph-nil}
The graph $\Gam _{\nu }(G)$ (resp. 
$\Gam ^+_{\nu }(G)$) is called the nilpotency graph 
(resp. directed nilpotency graph) of $G$. The diameters 
of these graphs are denoted by $d_{\nu }(G)$ and 
$d^+_{\nu }(G)$, respectively. (If there are vertices 
$g,h$ with no path from $g$ to $h$, the diameter is 
defined to be infinite.) 
\end{df}

\begin{prob} To investigate nilpotency graphs of finite 
simple groups and to estimate their diameters. 
\end{prob}


We turn now to ``solvability relations''. Here we 
immediately encounter the following 

\begin{prob} \label{prob:solv}
Find a sequence of words $\{e_n(x,y)\}_{n=1}^{\infty }$ in 
two variables $x,y$ such that a finite group $G$ is 
solvable if and only if for some $n$ the identity 
$e_n\equiv 1$ holds in $G$. 
\end{prob}

\noindent 
{\it Remark}. Similarly to the nilpotency case, we 
require that the identity $e_k\equiv 1$ would imply 
$e_l\equiv 1$ for all $l>k$. 

\bigskip

With such a sequence at our disposal, we could define 
the solvability relation in $G$ and the corresponding graphs 
repeating, word by word, Definitions \ref{def:rel-nil}--\ref{def:graph-nil}.  
 
There is strong evidence that Problem \ref{prob:solv} 
has a positive solution, and finite solvable groups can 
be characterized by two-variable identities. 

\begin{thm} \label{thm:Th} {\rm {\cite{T}, \cite{F}}}  
Let $G$ be a finite group in which every two elements 
generate a solvable subgroup. Then $G$ is solvable. 
\end{thm}

However, Theorem \ref{thm:Th} does not provide any 
explicit two-variable laws for finite solvable groups. 
We present several candidates for such $e_n$'s. 

\begin{df} \label{def:seq}
Let $\{ e_n(x,y)\}^{\infty }_{n=1}$ be defined by one of 
the following three formulae:

\begin{equation} \label{seq1}
\begin{array}{llllllllll}
e_1 & = & [x,y], & & & & & & & \\
e'_1 & = & [e_1,x], & e''_1 & = & [e_1,y], & e_2 & = & [e'_1,e''_1], 
& \dots \\ 
e'_n & = & [e_n,x], & e''_n & = & [e_n,y], & e_{n+1} & = 
& [e'_n,e''_n], & \dots
\end{array} 
\end{equation}


\begin{equation} \label{seq2}
e_1=[x,y], e_2=[xe_1x\min ,ye_1y\min ], \dots , 
e_{n+1}=[xe_nx\min ,ye_ny\min ], \dots 
\end{equation}


\begin{equation} \label{seq3}
\begin{array}{ll}
e'_1=x,  e''_1=y, e_1=[e'_1,e''_1],  \dots & \\ 
e'_{n+1}=[[e'_n,e''_n],e'_n],e''_{n+1}=[[e''_n,e'_n],e''_n],  
e_{n+1}=[e'_{n+1},e''_{n+1}], \dots 
\end{array}
\end{equation}

We call sequences $(\ref{seq1})$--$(\ref{seq3})$ reasonable.
\end{df}

Note that for all reasonable sequences, $e_k\equiv 1$ implies 
$e_l\equiv 1$ for all $l>k$. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\bigskip

\noindent
{\it Remark}. There is a natural way to produce reasonable 
sequences generalizing (\ref{seq2}). Namely, let $w$ be a word in 
$x,y,x\min ,y\min$. Define 
$$
e^w_1=w, e^w_{n+1}=[xe^w_nx\min ,ye^w_ny\min ],\dots .
$$
A clever choice of $w$ might lead to a sequence with good 
properties. We shall discuss the matter in detail in our 
forthcoming paper.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{df} 
Fix a reasonable sequence $\{e_i\}$. Let $G$ be 
an arbitrary group. We say that elements $g,h\in G$ are in 
relation $\sig _n$ (with respect to $\{e_i\}$) if $e_n(g,h)=1$. 
Elements $g,h$ are said 
to be in solvability relation $\sig$ if and only if they are 
in relation $\sig _n$ for some $n$. We call $\sig$-graph 
$\Gam _{\sig }(G)$ (resp. $\Gam ^+_{\sig }(G)$) the 
solvability graph (resp. the directed solvability graph) of 
$G$.
\end{df}

To justify the above definition, one has to prove the 
following analogue of the Engel property.

\begin{conj} {\text{\rm{(B. Plotkin)}}} \label{conj:main}
Let $\{e_i\}$ be a reasonable sequence. 
A finite group $G$ is solvable if and only if for some 
$n$ the identity $e_n\equiv 1$ holds in $G$.
\end{conj}

Clearly, if $G$ is solvable of derived length $n$, then $e_n\equiv 1$ 
holds in $G$. 


\medskip


\noindent
{\it Remark}. There is another way to define nilpotency 
and solvability relations: $g,h\in G$ are in relation $\nu$ 
(resp. $\sig$) if and only if the subgroup of $G$ generated 
by $g$ and $h$ is nilpotent (resp. solvable). Such relations  
have an advantage to be symmetric and disadvantage to be 
less constructive. We do not consider them in the present paper. 
We refer the reader to \cite{PPT} for yet another definition of 
the solvability graph and relationship with abstract algebraic geometry 
over groups. 
\bigskip

There are several results concerning characterization of solvable 
groups in terms of two-variable identities \cite{Ni1}, \cite{Ni2}, 
\cite{BN}. 
Namely, it was proved in \cite{Ni1}, \cite{Ni2} that if a finite 
group $G$ satisfies for 
some $n$ the identity $v_2\equiv v_n$, where $\{v_i\}$ is the 
sequence of Engel words, then $G$ is solvable. However, it is 
easy to find a solvable group satisfying no identity of the 
form $v_2\equiv v_n$. For example, take $G$ a finite nilpotent 
group of class $3$ such that the identity $v_2\equiv 1$ does 
not hold in $G$. Since $v_3\equiv 1$, the group $G$ cannot 
satisfy any identity of the form $v_2\equiv v_m$. However, 
$G$ is solvable. 

In \cite{BN} it was proved that the identity $v_3\equiv v_n$ 
can hold in certain finite simple groups such as $\PSL (2,4)$, 
$\PSL (2,8)$, etc. 
Let us also mention a pioneer result of N.~Gupta \cite{G}: 
any finite group satisfying the identity $v_1\equiv v_n$ is abelian. 
 
At this point, let us introduce some more new invariants 
of finite groups. Our first remark is that given any 
infinite sequence of distinct words in $m$ variables 
$\{w_i(x_1,\dots ,x_m)\}_{i=1}^{\infty }$, any finite group 
$G$ satisfies a law of the type 
$$
w_k(x_1,\dots ,x_m)\equiv w_l(x_1,\dots ,x_m).
$$
for some $k$ and $l$. (Indeed, the set of values of the $w_i$'s is finite.)  

The next definition goes back to \cite{GH}. It generalizes the notion of 
Engel depth (cf. \cite{Br}, \cite{BN}).

\begin{df} \label{def:inv-Engel}
The pair $(k,l)$ for which the identity 
$v_k(x,y)\equiv v_l(x,y)$ holds, with minimal $k+l$, 
is called the Engel invariant of $G$.        
\end{df}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\noindent
{\it Remark}. To justify the above definition, one has to 
check that the pair $(k,l)$ with the required properties 
is unique. Indeed, 
suppose that we have two Engel identities in $G$: 
$e_k\equiv e_l$ and $e_m\equiv e_n$ with $k<m<n<l$ and 
$k+l=m+n$ minimal with this property. 

We have $e_m(x,y)=e_n(x,y)$ for all $x,y$. Plug 
$e_{l-n}(x,y)$ instead of $x$. We get 
$e_{l-n+m}(x,y)=e_l(x,y)$ for all $x,y$. Hence 
$e_k(x,y)=e_{l-n+m}(x,y)$ for all $x,y$. 
Therefore, because of minimality of $k+l$, we have 
$k+(l-n+m)\geq k+l$, i.e. $m-n\geq 0$, contradiction.  
The Engel invariant is thus well defined. 

This remark also shows that the number $k$ in Definition 
\ref{def:inv-Engel} coincides with the Engel depth of $G$ 
as defined in \cite{Br}, \cite{BN}. However, the second 
parameter $l$ is also important as the following beautiful 
result shows \cite[Th.~4.3]{GH}: with the notation of 
Definition \ref{def:inv-Engel}, if $k+l$ is odd then 
$G$ is solvable. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{prob} \label{prob:Engel}
To compute Engel invariants for particular classes 
of finite groups.
\end{prob}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


In \cite{Nth}, Engel invariants have been computed in 
some groups, classes of groups and varieties of groups such as 
some groups of small order; the class of dihedral groups $D_p$ where $p$ 
is an odd prime; the solvable locally finite varieties of groups 
$A_kA_l$ for $k$ and $l$ powers of one and the same prime number $p$, 
and for $k$ and $l$ coprime integers; the infinite series of simple 
groups (the alternating groups $A_n$ for $n>5$ and the special projective 
groups $\PSL (2,q)$ for some of the first groups in the series). 

One can consider the analogues of Definition \ref{def:inv-Engel} 
and Problem \ref{prob:Engel} with the Engel sequence replaced by 
one of reasonable (in the sense of Definition \ref{def:seq}) 
sequences. 

\begin{df} \label{def:inv-sig}
Let $\{e_i\}$ be a reasonable sequence. 
The pair $(k,l)$ for which the identity 
$e_k(x,y)\equiv e_l(x,y)$ holds, with minimal $k+l$, 
is called $\sig$-invariant of $G$. If there are several 
such pairs, we choose among them the pair with 
minimal $k$ and define it to be $\sig (G)$.         
\end{df}



\begin{prob} 
To compute $\sig$-invariants for particular classes 
of finite groups.
\end{prob}
 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Lie-algebraic analogues}  \label{sec:Lie}

On replacing commutators by Lie products (and 1 by 0), we get 
sequences similar to (\ref{seq1})--(\ref{seq3}) for 
Lie algebras. We also call them reasonable. Here the situation 
is much more clear (at least, in finite dimensional 
case). We restrict ourselves by considering the Lie analogue 
of sequence (\ref{seq1}). 

\begin{thm} \label{prop:Lie-solv}
Let $L$ be a finite dimensional Lie algebra over an infinite field 
$k$ of characteristic $p>5$. Let $\{e_i\}$ be 
defined by formulae (\ref{seq1}). Then $L$ is solvable if and only if 
for some $n$ the identity $e_n\equiv 0$ holds in $L$. 
\end{thm}
{{\it Proof.}} 
Obviously, if $L$ is solvable then it satisfies 
an identity of the form $e_n(x,y)\equiv 0$ since for any 
$X,Y\in L$ the value $e_n(X,Y)$ belongs to the corresponding 
term of the derived series. Conversely, let $L$ 
satisfy the identity $e_n\equiv 0$. 
First suppose that $k$ is 
algebraically closed. 
If $L$ is not solvable then 
$L\ss =L/\rad (L)$ is semisimple and non-zero 
(here $\rad (L)$ denotes the solvable radical of $L$, i.e. 
its maximal solvable ideal). 
If $\char (F)=0$, let us  denote by 
$\{E_{\alp }, H_{\alp }, E_{-\alp }\}$ the standard basis 
of $sl_2$. Then $[E_{\alp },E_{-\alp }]=H_{\alp }$, 
$[H_{\alp },E_{\alp }]=2E_{\alp }$, 
$[H_{\alp },E_{-\alp }]=-2E_{-\alp }$. 
Set $x=E_{\alp }$, $y=E_{-\alp }$. Then 
$$
\begin{array}{ccccccccc}
e_1 & = & H_{\alp }, & e'_1 & = & 2E_{\alp }, & e''_1 & = & 
-2E_{-\alp }, \\
e_2 & = & -4H_{\alp }, & \dots , & & & & &  
\end{array}
$$
i.e. $e_n=mH_{\alp }$ with $m\neq 0$. Thus for any $n$ we 
have $e_n(E_{\alp }, E_{-\alp })\neq 0$. 

Let now $\char (k)=p>5$. 
First assume that $L$ is restricted (see \cite[2.1]{StF} 
for the definition; we refer to the same book for all 
background material in modular Lie algebras). Then we can use 
the classification theorem of \cite{BlW} in order to mimic 
the proof in characteristic zero. To be more precise, 
\cite{BlW} says that all simple restricted Lie algebras are given by 
the list predicted by the Kostrikin--Shafarevich conjecture. 
One can then verify that each of such algebras contains $sl_2$. 
For the algebras of classical type this is obvious. As to the 
algebras of Cartan type, one has to consider them as graded Lie 
algebras (see \cite[Ch.~4]{StF}) and notice that the zero 
component $L_0$ contains $sl_2$. Indeed, $S(n;{\bf 1})_0\cong sl_n$ 
(\cite[Prop.~3.3.4]{StF}), $H(2r;{\bf 1})_0\cong sp_{2r}$ 
(\cite[Prop.~4.4.4]{StF}), $K(2r+1;{\bf 1})_0$ contains $sp_{2r}$ 
(\cite[Ex.~4.5.3]{StF}), and $W(n;{\bf 1})_0\cong gl_n$ 
(\cite[Prop.~2.2.4]{StF}). In this last case for $n=1$ one has 
to consider the algebra $L_{-1}\oplus L_0\oplus L_1$; one can show 
that it is isomorphic to $sl_2$. 

If $L$ is not restricted, one needs more subtle arguments. 

\begin{lem} \label{lem}
Every simple Lie algebra $L$ defined over a field of characteristic 
$p>5$ contains a subalgebra $S$ with quotient isomorphic to $sl_2$. 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\footnote{A.~Premet informed us that one can modify the 
proof to be valid for all $p>2$.} 
%Namely, for such $p$ one can 
%show that every finite dimensional simple Lie algebra $L$ 
%contains a subalgebra $S$ with $S/\rad (S)$ isomorphic to 
%$sl_2$ (here $\rad (S)$ stands for the radical of $S$). 
%Since any identity remains true after going over to subalgebras 
%and factor-algebras, we are done.} 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{lem}

{\it Proof}. Assume the contrary. Let $L$ denote a counter-example 
of minimal dimension to the assertion of the lemma. Let $L^{\circ}$ 
denote a maximal subalgebra of $L$. We wish to show that $L^{\circ }$ 
is solvable. If not, then $L\ss = L^{\circ }/\rad (L^{\circ })$ is 
a non-zero semisimple Lie algebra. By \cite[Th.~9.3]{Bl}, 
$L\ss$ contains a simple algebra $S$. Let $S_1=\pi\min (S)$ be the 
preimage of $S$ with respect to the natural projection 
$\pi\colon L^{\circ }\to L\ss$, we have $S_1/\ker\pi\cong S$. Since 
$\dim S<\dim L$, there is a subalgebra $T\subseteq S$ and an ideal 
$J$ in $T$ such that $T/J\cong sl_2$. Denote $T_1=\pi\min (T)$, 
$J_1=\pi\min (J)$. We have $T_1/J_1\cong sl _2$, contradiction. 
We thus proved that $L^{\circ }$ is solvable. By \cite[Cor.~1.4]{W}, 
$L$ must be isomorphic either to $sl _2$, or to the Zassenhaus 
algebra $W(1;m)$. In the first case we are done. In the second case 
$L$ is graded, and we set $S=L_{-1}\oplus L_0\oplus L_1$. Each of the three 
components is one-dimensional, and a straightforward computation using the 
table of structure constants shows that $S\cong sl_2$. The lemma is 
proved. 
$\Box$

\medskip

Let us continue the proof of the theorem. We have a Lie algebra $L$ 
satisfying the identity $e_n(x,y)\equiv 0$. We wish to prove that 
$L$ is solvable. Assume the contrary. Then, arguing as in the proof of 
Lemma \ref{lem} (i.e. considering $L/\rad (L)$ and using 
\cite[Th.~9.3]{Bl}), we conclude that $L$ has a simple subalgebra $S$. 
From Lemma \ref{lem} it follows that $S$ (and hence $L$) has a subfactor 
isomorphic to 
$sl_2$. Since identities remain true in sub- and factor-algebras, 
$e_n\equiv 0$ must hold in $sl_2$, contradiction.  
We thus proved the ``if'' part 
of the theorem in the case where $k$ is algebraically closed. 

Let now $k$ be an arbitrary infinite field, and suppose that 
$L$ is a Lie algebra over $k$ satisfying an identity $w(x,y)\equiv 0$, 
where $w$ stands for one of the $e_n$'s.  We wish to prove that 
the identity $w(x,y)\equiv 0$ also holds in the Lie algebra 
$\Lb =L\otimes _k\kb$ defined over an algebraic closure 
$\kb$ of $k$. 
Indeed, let $\{E_1,\dots ,E_d\}$ denote a $k$-basis 
of $L$. On writing arbitrary $x,y\in L$ with respect to this basis: 
$x=\sum\alp _iE_i$, $y=\sum\bet _iE_i$,   
we translate the identity $w(x,y)\equiv 0$ into identities 
of the form 
$$
P_i(\alp _1,\dots ,\alp _d,\bet _1, \dots ,\bet _d)=0, \qquad i=1,\dots ,d,
$$
where $P_i$ are polynomials. Since all the values of each $P_i$ are zero 
and $k$ is infinite, we conclude that the $P_i$'s are zero polynomials 
(see, for example, \cite[Ch.~IV, \S 1, Cor.~1.7]{L}).   
Let now $\xb ,\yb\in\Lb$ are arbitrary elements. On writing them 
with respect to the same basis $\{E_i\}$ (with coefficients from $\kb$) 
and plugging into the expression for $w(\xb ,\yb )$, we obtain the 
same polynomials $P_i$ as coefficients at $E_i$. But we have already proved 
that they are zero. Hence $w(\xb ,\yb )=0$. 

%It is enough to prove that the identity remains true after 
%finite extension of $F/k$. But the Lie algebra $L\otimes F$, viewed 
%as a Lie algebra over $k$, is isomorphic to a direct sum of 
%$n$ copies of $L$, where $n=[F:k]$. The identity is true in the 
%direct sum because it is true in every component. 

The theorem is proved.
$\Box$
\medskip

Note one more result in the same spirit (cf. \cite{G}, 
\cite{GH}, \cite{Ni1}, \cite{Ni2}, \cite{BN} for the group case). 
Recall that $\{v_i\}$ is the Engel sequence, 
$v_i(x,y)=[[[x,y],y]\dots y]$ (for brevity we denote this expression 
by $[x,y_i]$).  

\begin{prop} \label{prop:Lie-Engel}
Let $L$ be a finite dimensional Lie algebra over a field 
of characteristic different from $2$ such that the 
identity $v_k\equiv v_l$  holds in $L$. Then $L$ is solvable. 
Moreover, if $L$ satisfies $v_1\equiv v_l$ then $L$ is abelian. 
\end{prop} 

{{\it Proof.} As in the preceding theorem, we can reduce to 
the case where the ground field is algebraically closed. First 
consider the characteristic zero case. Again, if $L$ is 
not solvable then $L\ss =L/\rad (L)$ is non-zero and contains 
$sl_2$. Set $x=H_{\alp }$, $y=E_{\alp }$. We have 
$v_1=2E_{\alp }$, $v_2=4E_{\alp }$, \dots , thus 
$v_k\equiv v_l$ leads to $2^kE_{\alp }=2^lE_{\alp }$, 
contradiction. In the positive characteristic, we 
just reproduce the arguments from Theorem \ref{prop:Lie-solv}.

Let now $v_1\equiv v_l$. We proceed by induction on $\dim L$. 
If $\dim L=1$ then $L$ is abelian. Suppose that all subalgebras 
of dimension less than $n=\dim L$ are abelian. By the first 
part of the proposition, $L$ is solvable. Hence $L'=[L,L]$ 
is of dimension less than $n$ and therefore is abelian. We have 
to prove that $L'=0$. Take any $[x,z]\in L'$. By assumption, 
$[x,z]=[x,z_l]$. Let $z=[x,y]$. Then $[x,[x,y]]=[x,[x,y]_l]=0$ 
since $L'$ is abelian. Therefore $[[x,y],x]=0$ and hence 
$[[y,x],x]=0$ and $[y,x_l]=0$. Applying our assumption once again 
we get $[y,x]=0$, so that $L'=0$ and thus $L$ is abelian.
$\Box$
\medskip

\noindent
{\it Remark}. As in the case of finite groups, one can 
note that the identity $v_2\equiv v_l$ gives only a sufficient 
condition for a finite dimensional Lie algebra to be solvable. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Identities in linear groups} \label{sec:lin}

One of the most promising approaches to the proof of 
Conjecture \ref{conj:main} is related to the study of 
identities in finite linear groups. To be more precise, 
the following question seems to be critical: let $\{e_i\}$ 
denote one of the reasonable sequences (see formulae 
(\ref{seq1})--(\ref{seq3})), and let $G=\PSL (2,p)$, $p>3$; 
is it true that neither of the formulae $e_n$ 
is an identity in $G$? (See the next section for more 
details.) It is known \cite{OP}, \cite[Cor. 52.12]{Ne} that any finite 
group $G$ has a finite basis       
of identities but for $\PSL (2,p)$ the explicit bases are known 
only for $p\leq 13$ (see \cite{CMS} and references therein). 
Clearly the identities 
of $\PSL (2,p)$ heavily depend on $p$ because $\PSL (2,\Z )$ 
has no identities at all. Thus looking at $\bG =\bPSL (2,\cdot )$ 
as at a group scheme, one can say that different values of $\bG$ 
have different identities. On the other hand, if an affine  
group scheme $\bG$ is assumed to be either abelian, 
or nilpotent, or solvable, then all its values inherit the 
corresponding identities. Therefore, given a linear group 
$G\subset\GL (n,k)$ isomorphic to the group $\bG (k)$ of $k$-rational 
points of an affine group scheme $\bG$,  it is important to 
distinguish its ``structural'' identities (i.e. coming from $\bG$) 
from those arising from the special choice of $k$. 

We now make all above considerations more precise. First 
introduce some notation. Given an affine group scheme $\bG$, 
we denote 

\begin{itemize}

\item $\mu\colon  \bG\times\bG\to\bG$, {\text{ \rm multiplication}}, 

\item $i\colon    \bG\to\bG$,  {\text{ \rm inversion}}, 

\item $e\colon    \bE\to\bG$, {\text{ \rm unit (where $\bE =\{e\}$ is the final 
object in the category of affine group schemes)}}, 

\item $c\colon    \bG\to\bG$, {\text{ \rm constant morphism, $c(g)=e$, 
(i.e. $c\colon\bG\to\bE\stackrel{e}{\to }\bG$, where 
$\bG\to\bE$ is the unique morphism from $\bG$ to $\bE$)}}, 

\item $\id\colon  \bG\to\bG$, {\text{ \rm identity}}, 

\item $t\colon    \bG\times\bG\to\bG\times\bG$, {\text{\rm transposition, i.e.  
$t=(\pr _2,\pr _1)$}}.

\end{itemize}

We wish to define the commutator $u\colon\bG\times\bG\to\bG$. 
Let $\tilde\mu\colon\bG\times\bG\times\bG\times\bG\to\bG$ be 
the composition $(\bG\times\bG )\times (\bG\times\bG )
\stackrel{\mu\times\mu }{\longrightarrow } \bG\times\bG\stackrel{\mu }
{\to }\bG$. We then define $u$ as the composite morphism 
$u\colon\bG\times\bG \stackrel{(i\times i,\id\times\id )} {\longrightarrow } 
\bG\times\bG\times\bG\times\bG\stackrel{\tilde\mu }{\to }\bG$. 

\bigskip

\noindent
{\it Observation}. A group scheme $\bG$ is commutative if and 
only if $u=c$.

\bigskip

\noindent
{\it Remark}. Of course, one can express the commutativity 
condition without using commutators, just saying that 
$\mu\circ t=\mu$.

\bigskip

We now want to generalize the above construction. We define, 
by unduction,
\begin{equation}
\begin{array}{llll}
e_1 & = & u=[x,y]=x\min y\min xy, & \dots \\ 
e_{n+1} & = & [[e_n,x],[e_n,y]], & \dots 
\end{array}
\end{equation}
More formally, we first define $e'_n=[e_n,x]$ and 
$e''_n=[e_n,y]$ as follows:
\begin{equation}
\begin{array}{ccccccc}
e'_n & \colon & \bG\times\bG & 
\stackrel{(e_n,\pr _1)}{\longrightarrow } 
& \bG\times\bG & \stackrel{u}{\to } & \bG , \\
e''_n & \colon & \bG\times\bG & 
\stackrel{(e_n,\pr _2)}{\longrightarrow }
& \bG\times\bG & \stackrel{u}{\to } & \bG . 
\end{array}
\end{equation}
Then $e_{n+1}$ is defined as the composite morphism
$$
e_{n+1}\colon\bG\times\bG \stackrel{(e'_n,e''_n)}{\longrightarrow } 
\bG\times\bG\stackrel{u}{\to }\bG . 
$$
The two other reasonable sequences (see formulae 
(\ref{seq2})--(\ref{seq3})) can be defined in a similar way. 

\begin{prop} \label{prop:struct-solv}
Let $\bG$ be a connected affine algebraic group over 
a field $k$. Then $\bG$ is solvable if and only if 
$e_n=c$ for some $n\geq 1$. 
\end{prop}

\noindent 
{{\it Proof.} {\it Necessity}. We prove by induction that 
the image of $e_n$ lies in the $n$-th derived subgroup 
$D^n\bG$ of $\bG$. For $n=1$ this is obvious. Since $D^n\bG$ is 
a normal subgroup in $\bG$, by induction hypothesis $e'_n$ and 
$e''_n$ each map $\bG\times\bG$ into $D^n\bG$. Hence $e_{n+1}$ maps 
$\bG\times\bG$ into $D^{n+1}\bG$. 

{\it Sufficiency}. First note that the condition $e_n=c$ is equivalent 
to the fact that all groups $\bG (A)$, where $A$ is any $k$-algebra, 
satisfy the identity $e_n(x,y)\equiv 1$. This property is thus 
hereditary with respect to sub- and factor-groups. Suppose that 
$\bG$ satisfies $e_n=c$ but is not solvable. In view of the above 
remark, the quotient $\bG\red =\bG /\bG\un$, where $\bG\un$ stands 
for the unipotent radical of $\bG$, is a non-trivial reductive 
group satisfying $e_n=c$. Furthermore, its derived group 
$\bG\ss=[\bG\red ,\bG\red ]$ is a non-trivial semisimple group 
satisfying the same property. Hence the $k$-group $\bSL _2$ being 
a subgroup of $\bG\ss$ must satisfy the same law. Its quotient 
$\bPSL _2$ thus also satisfies $e_n=c$. Therefore, the identity 
$e_n(x,y)\equiv 1$ must hold in all groups $\PSL _2(A)$ where 
$A$ is a $k$-algebra, that is impossible \cite{J}.      
$\Box$
\medskip

We now go over to a ``structural'' analogue of the Engel law. 
Define $v_1=u$ and, by induction, 
$$
v_{n+1}\colon\bG\times\bG \stackrel{(v_n,\pr _2)}{\longrightarrow } 
\bG\times\bG\stackrel{u}{\to }\bG.
$$
  
\begin{prop} \label{prop:struct-Ebgel} 
Let $\bG$ be a connected affine algebraic group over 
a field $k$. Then $\bG$ is nilpotent if and only if 
$v_n=c$ for some $n\geq 1$. 
\end{prop}
    
{{\it Proof.} Let $C^n\bG$ denote the $n$-th term of the 
lower central series. Then $v_n$ maps $\bG\times\bG$ into 
$C^n\bG$. This proves the ``only if'' part. Let now $v_n=c$. 
Then, as in the proof of Proposition \ref{prop:struct-solv}, 
we conclude that $\bG$ is solvable. Hence $\overline G=
\bG (\kb )$ is solvable (here $\kb$ stands for a (fixed) 
algebraic closure of $k$). According to \cite[IV, 4.1.5]{DG}, 
we only need to prove that $\overline G$ is nilpotent. Since 
$\overline G$ is solvable and connected, it is isomorphic 
to a semi-direct product $T\ltimes U$ where $T$ is a torus and 
$U$ is nilpotent. If $T=\{e\}$ or $U=\{e\}$ then $\overline G$ 
is nilpotent. Hence we may assume that $\dim T\geq 1$ and  
$\dim U\geq 1$. If $T$ is central in $\overline G$ then 
$\overline G$ is nilpotent \cite[10.6(3)]{Bo}. If not, 
$U$ contains a one-dimensional subgroup $U_1$ that does not 
commute with $T$. Since $\kb$ is algebraically closed, $U_1$ 
is isomorphic to the additive group ${\bf G}_{\rm a}$. Hence 
$T$ acts on $U_1$ as follows: $t\min ut=\lam (t)u$ where 
$u\in U_1$, $t\in T$, $\lam\colon T\to{\bf G}_{\rm m}$ is 
a character of $T$ (cf. \cite[10.10]{Bo}). Thus 
$t\min u\min t=\lam\min u\min$ and $t\min u\min tu=\lam\min$, 
so $[u,t]=\lam (t)\neq 1$ for $u\neq 1$, $t\neq 1$. By induction, 
we obtain $v_n(u,t)=\lam\neq 1$ for any $n$, which is a contradiction. 
$\Box$
\medskip

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Main conjecture} \label{sec:conj} 

In this section we return to Conjecture \ref{conj:main}. 
Our first observation is that by standard arguments one 
can reduce to considering only a finite number of series 
of finite simple groups. To be more precise, one can easily 
derive Conjecture \ref{conj:main} from the following 

\begin{conj} \label{conj:main'}
Let $G$ be one of the following groups: 
\begin{enumerate}

\item $\PSL (2,p)$ $(p=5$ {\text{ \rm or}} $p\equiv \pm 2\pmod 5, 
p\neq 3)$, 

\item $\PSL (2,2^p)$, 

\item $\PSL (2,3^p)$  $(p$ {\text{ \rm odd}}$)$, 

\item $\Sz (2^p)$  $(p$ {\text{ \rm odd}}$)$, 

\item $\PSL(3,3)$.
\end{enumerate}

Let $\{e_i\}$ be one of sequences $(\ref{seq1})$--$(\ref{seq3})$. 
Then neither of identities $e_n\equiv 1$ holds in $G$. 
\end{conj} 

Indeed, according to \cite{T}, the list of Conjecture 
\ref{conj:main'} is exactly the list of minimal finite non-solvable 
groups (that is, the groups whose every subgroup is solvable). 
On our way to proving Conjecture \ref{conj:main'}, we proceed 
by case-by-case computer investigation. In order to prove that 
$e_n\equiv 1$ is not a law in $G$, it is enough to show that 
for some $k<l$ the equation $e_k(x,y)=e_l(x,y)$ has a non-trivial 
solution $(x_0,y_0)\in G\times G$ (non-trivial means that 
$e_k(x_0,y_0)=e_l(x_0,y_0)\neq 1$; by the construction of the 
sequences, it suffices to check that the right-hand side does 
not equal 1). 

The case $G=\PSL (3,3)$ is the easiest one. Say, if the $e_n$'s 
are taken from sequence (\ref{seq1}), the matrices 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
$$
x_0=\left(
\matrix{
0 & 0 & 1 \cr
0 & 1 & 1 \cr 
2 & 1 & 1
}
\right) , 
\qquad 
y_0=\left(
\matrix{
1 & 0 & 1 \cr
0 & 2 & 0 \cr 
0 & 0 & 2
}
\right) 
$$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
give a solution to the equation $e_{17}=e_{21}$, and thus 
$e_n\equiv 1$ is not a law in $\PSL (3,3)$. 

For $G=\PSL (2,p)$, computer search gives a solution 
to $e_2=e_4$ (where the $e_i$'s are taken from 
sequence (\ref{seq2})) for all $p<1000$ except for 
$p=293$ (as in chess, $e2-e4$ 
usually wins!). The equation $e_3=e_5$ has a solution 
in $\PSL (2,p)$ for all $p<1000$, and this result remains 
true for all $p<1500$ except, possibly, for $p=1163$ for which 
calculations take too much time. 

See Appendix for more details concerning numerical 
experiments. 

To conclude, we present the following model case 
that can be viewed as a testing ground for proving 
Conjecture \ref{conj:main'}.

\begin{prop} \label{prop:e_2}
Let the sequence $\{e_i\}$ be given by formulae 
(\ref{seq1}). If the identity $e_2\equiv 1$ holds 
in a finite group $G$ then $G$ is solvable. 
\end{prop}

{{\it Proof.} As above, it is enough to prove that 
the identity $e_2\equiv 1$ does not hold in the 
minimal non-solvable groups. For $G=\PSL (3,3)$ 
it is proved above. Let now $G=\PSL (2,q)$. Take 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
$$ 
x=\left(
\matrix{
1 & 1 \cr
0 & 1
}
\right) , \qquad
y=\left(
\matrix{
1 & 0 \cr
t & 1
}
\right) .
$$
with $t\neq 0$. 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Then $e_2(x,y)=A(t)$ can be viewed as a polynomial matrix 
in indeterminate $t$. Its entry $A_{1,2}$ equals 
$-2t^3(t+1)f(t)$, where $f(t)=t^8+t^7-t^6-4t^5-8t^4-5t^3+t^2+4t+2$. 
Since $A_{1,2}(t)$ can only vanish at $t=0$, $t=-1$, and at the roots 
of $f(t)$, we conclude that for odd $q\geq 11$ the identity 
$e_2\equiv 1$ cannot hold in $\PSL (2,q)$. For $q=5$ and $q=7$ 
we have $A_{1,2}(1)\neq 0$. Thus we proved the proposition for 
$G=\PSL (2,p)$ and $G=\PSL (2,3^p)$. 

Next consider the case $G=\PSL (2,2^p)$. Take 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
$$ 
x=\left(
\matrix{
0 & 1 \cr
1 & 1
}
\right) , \qquad
y=\left(
\matrix{
t & 1 \cr
1 & 0
}
\right) .
$$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
As above, we have $e_2(x,y)=B(t)$, a polynomial matrix 
in one variable $t$ running over the finite field ${\Bbb F}_q$, 
$q=2^p$, 
with $B_{1,2}=t(t^8+1)$. 
One can easily see that $B_{1,2}(t)$ cannot vanish at all 
$t\in {\Bbb F}_q$.  
%Indeed, for $q\geq 8$ the degree of $B_{1,2}$ is too 
%small, and for $q=4$ the value of $B_{1,2}$ at a primitive 
%element of $\bF _4$ is non-zero. 

It only remains to consider the case of Suzuki groups $G=\Sz (q)$. 
We recall (see, for example, \cite[Ch.~XI, \S 3]{HB}) that as a subgroup of 
$\GL (4,q)$ the group $G$ is generated by the following matrices: 
$$
S(\alp ,\bet )=\left(
\matrix{
1 & 0 & 0 & 0 \cr
\alp ^z & 1 & 0 & 0 \cr
\bet & \alp & 1 & 0 \cr
\alp ^{2z+1}+\alp ^z\beta +\bet ^{2z} & \alp ^{1+z}+\bet & \alp ^z & 1
}
\right) , \qquad \alp ,\bet\in{\Bbb F}_q, 
$$
$$
M(\zeta )=\left(
\matrix{
\zeta ^z & 0 & 0 & 0 \cr
0 & \zeta ^{1-z} & 0 & 0 \cr
0 & 0 & \zeta ^{z-1} & 0 \cr
0 & 0 & 0 & \zeta ^{-z} 
}
\right)  , \qquad \zeta\in{\Bbb F}_q^*,  
\qquad {\text{\rm and }} 
J=\left(
\matrix{
0 & 0 & 0 & 1 \cr
0 & 0 & 1 & 0 \cr
0 & 1 & 0 & 0 \cr
1 & 0 & 0 & 0 
}
\right) .
$$ 
Here  $z=2^{\frac{p-1}{2}}$. We now take $x=J$, $y=S(1,t)$.  
As above, a straightforward computation (using MAPLE) gives  
$e_2(x,y)$ as a polynomial matrix $C(t)$. Its entry 
$C_{4,1}(t)$ equals $t^{2z}(t^z+1)$. The number of roots 
of $t^z+1$ in ${\Bbb F}_q$ does not exceed $z$ which is strictly 
smaller than $q-1$, and thus $C_{4,1}(t)$ cannot vanish at all 
$t\in{\Bbb F}_q$.   
$\Box$

\bigskip

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\noindent   
{\it Remark}. Probably one can characterize neither finite nilpotent groups
of a fixed class, nor finite solvable groups of a fixed
derived length by means of two-variable identities. 
%Indeed, it is known \cite{Le} that every nilpotent group
%with $2$-generator subgroups of class 2 has nilpotency class $3$. 
For example, there exists a non-metabelian solvable  
group $G$ such that all its $2$-generator subgroups are metabelian
\cite{BHN}, \cite{CG}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section*{Appendix} \label{app}
We present here some results of computer experiments. In Table 1 
 for each $p<200$ we exhibit one solution $(x,y)$ to 
the equation $e_2(x,y)=e_4(x,y)$ where the $e_i$'s are taken 
from sequence (\ref{seq2}). In the next two tables for $p<80$ 
we present the number of solutions to the above equation for 
sequences (\ref{seq1}), (\ref{seq2}), respectively. 
Finally, we include the C++ code for the programs computing the 
number of solutions.

%\newpage

%\begin{table}[!h]
\small{

\begin{tabular}{|c|cc|c|cc|}
\hline
$p$ & $x$ & $y$ & $p$ & $x$ & $y$ \\
\hline
5 & 
$\left(
\matrix{
0 & -1 \\
1 & 1
}
\right)$
&
$\left(
\matrix{
-2 & -1 \\
1  & 0
}
\right)$
&
7 & 
$\left(
\matrix{
0 & -1 \\
1 &  3
}
\right)$
&
$\left( 
\matrix{
-3 & 2 \\
-2 & 1
}
\right)$ 
\\
\hline
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
11 & 
$\left(
\matrix{
0 & -1 \\
1 & 1
}
\right)$ 
&
$\left(
\matrix{
-5 & -5 \\
0  & 2
}
\right)$ 
&
13 & 
$\left(
\matrix{
0 & -1 \\
1 &  6
}
\right)$
&
$\left( 
\matrix{
-5 & -5 \\
-6 & -1
}
\right)$ 
\\
\hline
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
17 & 
$\left(
\matrix{
0 & -1 \\
1 & 1
}
\right)$ 
&
$\left(
\matrix{
-8 & -5 \\
3  & 6
}
\right)$ 
&
19 & 
$\left(
\matrix{
0 & -1 \\
1 &  1
}
\right)$
&
$\left( 
\matrix{
-9 & -8 \\
1  & 5
}
\right)$ 
\\
\hline
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
23 & 
$\left(
\matrix{
0 & -1 \\
1 & 11
}
\right)$ 
&
$\left(
\matrix{
-11 & -11 \\
-10  & -8
}
\right)$ 
&
29 & 
$\left(
\matrix{
0 & -1 \\
1 &  5
}
\right)$
&
$\left( 
\matrix{
-14 & -13 \\
-6 & 13
}
\right)$ 
\\
\hline
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
31 & 
$\left(
\matrix{
0 & -1 \\
1 & 5
}
\right)$ 
&
$\left(
\matrix{
-15 & -15 \\
-2  & 0
}
\right)$ 
&
37 & 
$\left(
\matrix{
0 & -1 \\
1 &  3
}
\right)$
&
$\left( 
\matrix{
-18 & -12 \\
11 & -3
}
\right)$ 
\\
\hline
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
41 & 
$\left(
\matrix{
0 & -1 \\
1 & 12
}
\right)$ 
&
$\left(
\matrix{
-20 & -14 \\
-3  & 4
}
\right)$ 
&
43 & 
$\left(
\matrix{
0 & -1 \\
1 &  1
}
\right)$
&
$\left( 
\matrix{
-21 & -20 \\
1   & 5
}
\right)$ 
\\
\hline
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
47 & 
$\left(
\matrix{
0 & -1 \\
1 & 16
}
\right)$ 
&
$\left(
\matrix{
-23 & -21 \\
-20  & 4
}
\right)$ 
&
53 & 
$\left(
\matrix{
0 & -1 \\
1 &  19
}
\right)$
&
$\left( 
\matrix{
-26 & -24 \\
-18 & 18
}
\right)$ 
\\
\hline
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
59 & 
$\left(
\matrix{
0 & -1 \\
1 & 24
}
\right)$ 
&
$\left(
\matrix{
-29 & -24 \\
23  & 19
}
\right)$ 
&
61 & 
$\left(
\matrix{
0 & -1 \\
1 &  29
}
\right)$
&
$\left( 
\matrix{
-30 & -28 \\
-12 & 3
}
\right)$ 
\\
\hline
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
67 & 
$\left(
\matrix{
0 & -1 \\
1 & 2
}
\right)$ 
&
$\left(
\matrix{
-33 & -28 \\
17  & -12
}
\right)$ 
&
71 & 
$\left(
\matrix{
0 & -1 \\
1 &  34
}
\right)$
&
$\left( 
\matrix{
-35 & -32 \\
-26 & 33
}
\right)$ 
\\
\hline
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
73 & 
$\left(
\matrix{
0 & -1 \\
1 & 22
}
\right)$ 
&
$\left(
\matrix{
-36 & -36 \\
12  & 14
}
\right)$ 
&
79 & 
$\left(
\matrix{
0 & -1 \\
1 &  29
}
\right)$
&
$\left( 
\matrix{
-39 & -39 \\
36 & 38
}
\right)$ 
\\
\hline
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
83 & 
$\left(
\matrix{
0 & -1 \\
1 & 34
}
\right)$ 
&
$\left(
\matrix{
-41 & -35 \\
-40  & -20
}
\right)$ 
&
89 & 
$\left(
\matrix{
0 & -1 \\
1 &  39
}
\right)$
&
$\left( 
\matrix{
-44 & -44 \\
38 & 40
}
\right)$ 
\\
\hline
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
97 & 
$\left(
\matrix{
0 & -1 \\
1 & 13
}
\right)$ 
&
$\left(
\matrix{
-48 & -45 \\
-3  & -19
}
\right)$ 
&
101 & 
$\left(
\matrix{
0 & -1 \\
1 & 45
}
\right)$
&
$\left( 
\matrix{
-50 & -45 \\
-17 & 17
}
\right)$ 
\\
\hline
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
103 & 
$\left(
\matrix{
0 & -1 \\
1 & 19
}
\right)$ 
&
$\left(
\matrix{
-51 & -44 \\
8  & 19
}
\right)$ 
&
107 & 
$\left(
\matrix{
0 & -1 \\
1 &  47
}
\right)$
&
$\left( 
\matrix{
-53 & -31 \\
-1 & -43
}
\right)$ 
\\
\hline
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
109 & 
$\left(
\matrix{
0 & -1 \\
1 & 1
}
\right)$ 
&
$\left(
\matrix{
-54 & -49 \\
5  & -52
}
\right)$ 
&
113 & 
$\left(
\matrix{
0 & -1 \\
1 & 47
}
\right)$
&
$\left( 
\matrix{
-56 & -56 \\
-11 & -9
}
\right)$ 
\\
\hline
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
127 & 
$\left(
\matrix{
0 & -1 \\
1 & 8
}
\right)$ 
&
$\left(
\matrix{
-63 & -55 \\
-11  & -58
}
\right)$ 
&
131 & 
$\left(
\matrix{
0 & -1 \\
1 &  11
}
\right)$
&
$\left( 
\matrix{
-65 & -58 \\
13 & -65
}
\right)$ 
\\
\hline
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
137 & 
$\left(
\matrix{
0 & -1 \\
1 & 11
}
\right)$ 
&
$\left(
\matrix{
-68 & -54 \\
-59  & -65
}
\right)$ 
&
139 & 
$\left(
\matrix{
0 & -1 \\
1 &  1
}
\right)$
&
$\left( 
\matrix{
-69 & -68 \\
1 & 5
}
\right)$ 
\\
\hline
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
149 & 
$\left(
\matrix{
0 & -1 \\
1 & 40
}
\right)$ 
&
$\left(
\matrix{
-74 & -69 \\
20  & 73
}
\right)$ 
&
151 & 
$\left(
\matrix{
0 & -1 \\
1 & 64
}
\right)$
&
$\left( 
\matrix{
-75 & -73 \\
-21 & 48
}
\right)$ 
\\
\hline
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
157 & 
$\left(
\matrix{
0 & -1 \\
1 & 1
}
\right)$ 
&
$\left(
\matrix{
-78 & -68 \\
10  & 55
}
\right)$ 
&
163 & 
$\left(
\matrix{
0 & -1 \\
1 & 58
}
\right)$
&
$\left( 
\matrix{
-81 & -30 \\
-59 & -44
}
\right)$ 
\\
\hline
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
167 & 
$\left(
\matrix{
0 & -1 \\
1 & 14
}
\right)$ 
&
$\left(
\matrix{
-83 & -76 \\
-83  & -74
}
\right)$ 
&
173 & 
$\left(
\matrix{
0 & -1 \\
1 &  37
}
\right)$
&
$\left( 
\matrix{
-86 & -63 \\
-23 & -41
}
\right)$ 
\\
\hline
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
179 & 
$\left(
\matrix{
0 & -1 \\
1 & 53
}
\right)$ 
&
$\left(
\matrix{
-89 & -88 \\
-59  & 4
}
\right)$ 
&
181 & 
$\left(
\matrix{
0 & -1 \\
1 &  49
}
\right)$
&
$\left( 
\matrix{
-90 & -86 \\
-20 & 3
}
\right)$ 
\\
\hline
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
191 & 
$\left(
\matrix{
0 & -1 \\
1 & 88
}
\right)$ 
&
$\left(
\matrix{
-95 & -85 \\
-41  & -95
}
\right)$ 
&
193 & 
$\left(
\matrix{
0 & -1 \\
1 &  84
}
\right)$
&
$\left( 
\matrix{
-96 & -94 \\
-62 & 78
}
\right)$ 
\\
\hline
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
197 & 
$\left(
\matrix{
0 & -1 \\
1 & 24
}
\right)$ 
&
$\left(
\matrix{
-98 & -86 \\
-20  & 93
}
\right)$ 
&
199 & 
$\left(
\matrix{
0 & -1 \\
1 & 1
}
\right)$
&
$\left( 
\matrix{
-99 & -95 \\
4 & 38 
}
\right)$
\\ 
\hline
\end{tabular}
}

\bigskip

\centerline{{\sc Table 1}: Solutions to $e_2=e_4$ (formulae (\ref{seq1}))}

\bigskip

%\end{table}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

}

%\newpage




%\begin{table}[!h]
\begin{tabular}{|c|cccccccccc|}
\hline
$p$ & 5 & 7 & 11 & 13 & 17 & 19 & 23 & 29 & 31 & 37  \\
\hline
$N_1$ & 0 & 84 & 96 & 300 & 668 & 80 & 88 & 360 & 760 & 440 \\  
\hline
\hline
$p$ & 41 & 43 & 47 & 53 & 59 & 61 & 67 & 71 & 73 & 79 \\ 
\hline
$N_1$ & 664 & 848 & 1312 & 428 & 712 & 480 & 1616 & 1432 & 1168 & 1904 \\
\hline
\end{tabular}

\bigskip

\centerline{{\sc Table 2}: Numbers of solutions to $e_2=e_4$ 
(formulae (\ref{seq1}))}

\bigskip

%\end{table}

%%%%%%%%%%%%%%%%%%%%%%%


%\begin{table}[!h]
\begin{tabular}{|c|cccccccccc|}
\hline
$p$ & 5 & 7 & 11 & 13 & 17 & 19 & 23 & 29 & 31 & 37  \\
\hline
$N_2$ & 22 & 16 & 134 & 28 & 36 & 304 & 136 & 526 & 670 & 296 \\  
\hline
\hline
$p$ & 41 & 43 & 47 & 53 & 59 & 61 & 67 & 71 & 73 & 79 \\ 
\hline
$N_2$ & 990 & 590 & 760 & 428 & 1064 & 728 & 402 & 1136 & 584 & 2050 \\
\hline
\end{tabular}

\bigskip

\centerline{{\sc Table 3}: Numbers of solutions to $e_2=e_4$ 
(formulae (\ref{seq2}))}
%\end{table}

\bigskip

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{verbatim}
//#==============================================================#
//#                                                              #
//#       The sequence U1 = [X,Y]; Un+1 = [[Un,X],[Un,Y]]        #
//#                                                              #
//#                X = ||0 -1||  Y = ||a b||                     #
//#                    ||1  t||      ||c d||                     #
//#                                                              #
//#==============================================================#

//#==============================================================#
//#                                                              #
//# The program computes the number of the solutions for U2= U4  #
//#                                                   or U2=-U4  #
//#               for given prime p                              #
//#                                                              #
//#==============================================================#


#include <stdio.h>
#include <math.h>
#include <stdlib.h>

//***************************************************
void display(long data[][2]);
void multiply(long X[2][2],long Y[2][2],int p);
void inv_X(long X[2][2]);
void inv_Y(long Y[2][2]);
void U_2(long X[2][2],long Y[2][2],int p);
void U_3(long X[2][2],long Y[2][2],int p);
void U_4(long X[2][2],long Y[2][2],int p);
void minus_U_4();
void Commutator(long X[2][2],long Y[2][2],int p);
int find_number_of_solutions(int p);
bool equal(long X[2][2],long Y[2][2],int p);


//***************************************************
//Global variables
static long C[2][2];
static long U2[2][2];
static long U4[2][2];
static long minus_U4[2][2];
static long U3[2][2];
static long M[2][2];
static long InvX[2][2];
static long InvY[2][2];
static long tmp1[2][2];
static long tmp2[2][2];
static long tmp[2][2];
static long temp1[2][2];
static long temp2[2][2];
FILE *output;
//***************************************************

int main(void)   {
int p ;

output = fopen("numbers.txt", "a+");
fclose(output);
printf("Enter the prime number P: ");
scanf("%d",&p);
printf("\n");
output = fopen("numbers.txt", "a+");
fprintf(output, "======");
fprintf(output, "P = ");
fprintf(output, "%d", p," ");
fprintf(output,"\n ");
fprintf(output, "The number of solutions is: ");
fprintf(output, "%d", find_number_of_solutions(p)," ");      
fprintf(output, "======");
fprintf(output,"\n ");
fclose(output);
 return(0);
}
//*****************************************************************************
void display(long data[2][2]) {
output = fopen("numbers.txt", "a+");
   fprintf(output, " \n*********************************\n");
   for (int i = 0; i < 2; i++) {
        for (int j = 0; j < 2; j++)
        {
             fprintf(output, "%d", data[i][j]);
             fprintf(output, " ");
          }
             fprintf(output, " \n");

       }
       fprintf(output, " \n*********************************\n");
       fclose(output);
   }
//******************************************************************************

void multiply(long X[2][2],long Y[2][2],int p)
{
 M[0][0] = (X[0][0]*Y[0][0]+X[0][1]*Y[1][0])%p;
 M[0][1] = (X[0][0]*Y[0][1]+X[0][1]*Y[1][1])%p;
 M[1][0] = (X[1][0]*Y[0][0]+X[1][1]*Y[1][0])%p;
 M[1][1] = (X[1][0]*Y[0][1]+X[1][1]*Y[1][1])%p;

}
//***************************************************

 void inv_X(long X[2][2])
 {
  InvX[0][0] =  X[1][1];
  InvX[0][1] = -X[0][1];
  InvX[1][0] = -X[1][0];
  InvX[1][1] =  X[0][0];
  }

  void inv_Y(long Y[2][2])
 {
  InvY[0][0] =  Y[1][1];
  InvY[0][1] = -Y[0][1];
  InvY[1][0] = -Y[1][0];
  InvY[1][1] =  Y[0][0];
  }
//******************************************************************************
  void Commutator(long X[2][2],long Y[2][2],int p)
  {
   inv_X(X);
   inv_Y(Y);
   multiply(InvX,InvY,p);
   for(int i=0;i<2;i++)
     for(int j=0;j<2;j++)
       temp1[i][j]=M[i][j];
   multiply(temp1,X,p);
   for( i=0;i<2;i++)
     for(int j=0;j<2;j++)
       temp2[i][j]=M[i][j];
   multiply(temp2,Y,p);
   for( i=0;i<2;i++)
     for(int j=0;j<2;j++)
     {
      C[i][j]=M[i][j];
      M[i][j]=0;
     } 
   }

//*****************************************************************************
  void U_2(long X[2][2],long Y[2][2],int p)
  {
   Commutator(X,Y,p);
    for(int i=0;i<2;i++)
     for(int j=0;j<2;j++)
      tmp[i][j]=C[i][j];
   Commutator(tmp,X,p);
    for( i=0;i<2;i++)
     for(int j=0;j<2;j++)
      tmp1[i][j]=C[i][j];
   Commutator(tmp,Y,p);
    for( i=0;i<2;i++)
     for(int j=0;j<2;j++)
      tmp2[i][j]=C[i][j];
     Commutator(tmp1,tmp2,p);
    for( i=0;i<2;i++)
     for(int j=0;j<2;j++)
      U2[i][j]=C[i][j];
    }

  void U_3(long X[2][2],long Y[2][2],int p)
  {
   Commutator(U2,X,p);
   for(int i=0;i<2;i++)
     for(int j=0;j<2;j++)
      tmp1[i][j]=C[i][j];
   Commutator(U2,Y,p);
   for(i=0;i<2;i++)
     for(int j=0;j<2;j++)
      tmp2[i][j]=C[i][j];

   Commutator(tmp1,tmp2,p);
   for(int i=0;i<2;i++)
     for(int j=0;j<2;j++)
      U3[i][j]=C[i][j];
 }

void U_4(long X[2][2],long Y[2][2],int p)
  {
   Commutator(U3,X,p);
   for(int i=0;i<2;i++)
     for(int j=0;j<2;j++)
      tmp1[i][j]=C[i][j];
   Commutator(U3,Y,p);
   for( i=0;i<2;i++)
     for(int j=0;j<2;j++)
      tmp2[i][j]=C[i][j];

   Commutator(tmp1,tmp2,p);
   for(int i=0;i<2;i++)
     for(int j=0;j<2;j++)
      U4[i][j]=C[i][j];
 }

void minus_U_4()
{
 for(int i=0;i<2;i++)
     for(int j=0;j<2;j++)
      minus_U4[i][j]=-U4[i][j];
}
 
//******************************************************************************
bool equal(long X[2][2],long Y[2][2],int p)
{
 if((((X[0][0]-Y[0][0])%p)==0)&&
    (((X[0][1]-Y[0][1])%p)==0)&&
    (((X[1][0]-Y[1][0])%p)==0)&&
    (((X[1][1]-Y[1][1])%p)==0))
    return(true);
  else
    return(false);
}

//******************************************************************************

int find_number_of_solutions(int p)
{
 static long E[2][2];
 E[0][0]=1;
 E[0][1]=0;
 E[1][0]=0;
 E[1][1]=1;
 static long X[2][2];
 static long Y[2][2];
 int counter =0;

 for(int i=(1-p)/2;i<=(p-1)/2;i++)
    for(int j=(1-p)/2;j<=(p-1)/2;j++)
		for(int m=(1-p)/2;m<=(p-1)/2;m++)
			for(int n=(1-p)/2;n<=(p-1)/2;n++)
				for(int k=(1-p)/2;k<=(p-1)/2;k++)
            {
					if (((i*n-j*m-1)%p)==0)
               {
                X[0][0]=0;
                X[0][1]=-1;
                X[1][0]=1;
                X[1][1]=k;

                Y[0][0]=i;
                Y[0][1]=j;
                Y[1][0]=m;
                Y[1][1]=n;

                U_2(X,Y,p);
                U_3(X,Y,p);
                U_4(X,Y,p);
		        minus_U_4(); 
               

					  if((!equal(U3,E,p))&&
						  ((equal(U2,U4,p))||
                           (equal(U2,minus_U4,p))))
					{
                        counter=counter+1; 
					  	
                    }
                   } 
                }
					return(counter);
}
\end{verbatim}

\centerline{{\sc Figure 1}: Program for computing the number of solutions 
to $e_2=e_4$ (formulae (\ref{seq1}))} 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{verbatim}

//#==============================================================#
//#                                                              #
//#    The sequence U1 = [X,Y]; Un+1 = [ X^-1*Un*X,Y^-1*Un*Y ]   #
//#                                                              #
//#                X = ||0 -1||  Y = ||a b||                     #
//#                    ||1  t||      ||c d||                     #
//#                                                              #
//#==============================================================#


//#==============================================================#
//#                                                              #
//# The program computes the number of the solutions for U2= U4  #
//#                                                   or U2=-U4  #
//#               for given prime p                              #
//#                                                              #
//#==============================================================#


#include <stdio.h>
#include <math.h>
#include <stdlib.h>

//***************************************************
void display(long  data[][2]);
void multiply(long  X[2][2],long  Y[2][2],int p);
void inv_X(long  X[2][2]);
void inv_Y(long  Y[2][2]);
void U_2(long  X[2][2],long  Y[2][2],int p);
void U_3(long  X[2][2],long  Y[2][2],int p);
void U_4(long  X[2][2],long  Y[2][2],int p);
void minus_U_4();
void Commutator(long  X[2][2],long  Y[2][2],int p);
int find_number_of_solutions(int p);
bool equal(long  X[2][2],long  Y[2][2],int p);


//***************************************************
//Global variables
static long  C[2][2];
static long  U2[2][2];
static long  U4[2][2];
static long  U3[2][2];
static long  minus_U4[2][2];
static long  M[2][2];
static long  InvX[2][2];
static long  InvY[2][2];
static long  tmp1[2][2];
static long  tmp2[2][2];
static long  tmp3[2][2];
static long  tmp4[2][2];
static long  tmp[2][2];
static long  temp1[2][2];
static long  temp2[2][2];
FILE *output;
//***************************************************

int main(void)   {
int p ;

output = fopen("numbers1.txt", "a+");
fclose(output);
printf("Enter the prime number P: ");
scanf("%d",&p);
printf("\n");
output = fopen("numbers1.txt", "a+");
fprintf(output, "======");
fprintf(output, "P = ");
fprintf(output, "%d", p," ");
fprintf(output,"\n ");
fprintf(output, "The number of solutions is: ");
fprintf(output, "%d", find_number_of_solutions(p)," ");
fprintf(output, "======");
fprintf(output,"\n ");
fclose(output);
 return(0);
}
//*****************************************************************************
void display(long  data[2][2]) {
output = fopen("numbers1.txt", "a+");
   fprintf(output, " \n*********************************\n");
   for (int i = 0; i < 2; i++) {
        for (int j = 0; j < 2; j++)
        {
             fprintf(output, "%d", data[i][j]);
             fprintf(output, " ");
          }
             fprintf(output, " \n");

       }
       fprintf(output, " \n*********************************\n");
       fclose(output);
   }
//******************************************************************************

void multiply(long  X[2][2],long  Y[2][2], int p)
{
 M[0][0] = (X[0][0]*Y[0][0]+X[0][1]*Y[1][0])%p;
 M[0][1] = (X[0][0]*Y[0][1]+X[0][1]*Y[1][1])%p;
 M[1][0] = (X[1][0]*Y[0][0]+X[1][1]*Y[1][0])%p;
 M[1][1] = (X[1][0]*Y[0][1]+X[1][1]*Y[1][1])%p;

}
//***************************************************

 void inv_X(long  X[2][2])
 {
  InvX[0][0] =  X[1][1];
  InvX[0][1] = -X[0][1];
  InvX[1][0] = -X[1][0];
  InvX[1][1] =  X[0][0];
  }

  void inv_Y(long  Y[2][2])
 {
  InvY[0][0] =  Y[1][1];
  InvY[0][1] = -Y[0][1];
  InvY[1][0] = -Y[1][0];
  InvY[1][1] =  Y[0][0];
  }
//******************************************************************************
  void Commutator(long  X[2][2],long  Y[2][2],int p)
  {
   //static int temp[2][2];
   //static long  temp1[2][2];
   //static long  temp2[2][2];
   inv_X(X);
   inv_Y(Y);
   multiply(InvX,InvY,p);
   for( int i=0;i<2;i++)
     for(int j=0;j<2;j++)
       temp1[i][j]=M[i][j];
   multiply(temp1,X,p);
   for( i=0;i<2;i++)
     for(int j=0;j<2;j++)
       temp2[i][j]=M[i][j];
   multiply(temp2,Y,p);
   for(i=0;i<2;i++)
     for(int j=0;j<2;j++)
     {
      C[i][j]=M[i][j];
      M[i][j]=0;
     } 
   }

//*****************************************************************************
  void U_2(long  X[2][2],long  Y[2][2],int p)
  {
   Commutator(X,Y,p);
    for(int i=0;i<2;i++)
     for(int j=0;j<2;j++)
      tmp[i][j]=C[i][j];
   multiply(InvX,tmp,p);
    for( i=0;i<2;i++)
     for(int j=0;j<2;j++)
      tmp1[i][j]=M[i][j];
   multiply(tmp1,X,p);
    for( i=0;i<2;i++)
     for(int j=0;j<2;j++)
      tmp2[i][j]=M[i][j];

    multiply(InvY,tmp,p);
    for( i=0;i<2;i++)
     for(int j=0;j<2;j++)
      tmp3[i][j]=M[i][j];
   multiply(tmp3,Y,p);
    for( i=0;i<2;i++)
     for(int j=0;j<2;j++)
      tmp4[i][j]=M[i][j];
    Commutator(tmp2,tmp4,p);
    for( i=0;i<2;i++)
     for(int j=0;j<2;j++)
      U2[i][j]=C[i][j];
    }

  void U_3(long  X[2][2],long  Y[2][2],int p)
  {
   inv_X(X);
   inv_Y(Y);
   multiply(InvX,U2,p);
    for(int i=0;i<2;i++)
     for(int j=0;j<2;j++)
      tmp1[i][j]=M[i][j];
   multiply(tmp1,X,p);
    for( i=0;i<2;i++)
     for(int j=0;j<2;j++)
      tmp2[i][j]=M[i][j];

    multiply(InvY,U2,p);
    for( i=0;i<2;i++)
     for(int j=0;j<2;j++)
      tmp3[i][j]=M[i][j];
   multiply(tmp3,Y,p);
    for( i=0;i<2;i++)
     for(int j=0;j<2;j++)
      tmp4[i][j]=M[i][j];
    Commutator(tmp2,tmp4,p);
    for( i=0;i<2;i++)
     for(int j=0;j<2;j++)
      U3[i][j]=C[i][j];
 }

void U_4(long  X[2][2],long  Y[2][2],int p)
  {
   inv_X(X);
   inv_Y(Y);
   multiply(InvX,U3,p);
    for(int i=0;i<2;i++)
     for(int j=0;j<2;j++)
      tmp1[i][j]=M[i][j];
   multiply(tmp1,X,p);
    for(i=0;i<2;i++)
     for(int j=0;j<2;j++)
      tmp2[i][j]=M[i][j];

    multiply(InvY,U3,p);
    for( i=0;i<2;i++)
     for(int j=0;j<2;j++)
      tmp3[i][j]=M[i][j];
   multiply(tmp3,Y,p);
    for( i=0;i<2;i++)
     for(int j=0;j<2;j++)
      tmp4[i][j]=M[i][j];
    Commutator(tmp2,tmp4,p);
    for( i=0;i<2;i++)
     for(int j=0;j<2;j++)
      U4[i][j]=C[i][j];
 }

 void minus_U_4()
  {
   for(int i=0;i<2;i++)
     for(int j=0;j<2;j++)
      minus_U4[i][j]=-U4[i][j];
 }
//******************************************************************************
bool equal(long  X[2][2],long  Y[2][2],int p)
{
 if((((X[0][0]-Y[0][0])%p)==0)&&
    (((X[0][1]-Y[0][1])%p)==0)&&
    (((X[1][0]-Y[1][0])%p)==0)&&
    (((X[1][1]-Y[1][1])%p)==0))
    return(true);
  else
    return(false);
}

//******************************************************************************

int find_number_of_solutions(int p)
{
 static long E[2][2];
 E[0][0]=1;
 E[0][1]=0;
 E[1][0]=0;
 E[1][1]=1;
 static long minus_E[2][2];
 minus_E[0][0]=-1;
 minus_E[0][1]=0;
 minus_E[1][0]=0;
 minus_E[1][1]=-1;
 static long  X[2][2];
 static long  Y[2][2];
 int counter=0;

 for(int i=(1-p)/2;i<=(p-1)/2;i++)
    for(int j=(1-p)/2;j<=(p-1)/2;j++)
		for(int m=(1-p)/2;m<=(p-1)/2;m++)
			for(int n=(1-p)/2;n<=(p-1)/2;n++)
				for(int k=1;k<=(p-1)/2;k++)
            {
					if (((i*n-j*m-1)%p)==0)
               {
                X[0][0]=0;
                X[0][1]=-1;
                X[1][0]=1;
                X[1][1]=k;

                Y[0][0]=i;
                Y[0][1]=j;
                Y[1][0]=m;
                Y[1][1]=n;

                U_2(X,Y,p);
                U_3(X,Y,p);
                U_4(X,Y,p);
                minus_U_4();

					  if(((!equal(U4,E,p))&&(!equal(U4,minus_E,p)))&&
						  ((equal(U2,U4,p))||
                     (equal(U2,minus_U4,p))))
					{
                        counter=counter+1;

                    }
                   }
                }
					return(counter);

}
\end{verbatim}

\centerline{{\sc Figure 2}: Program for computing the number of solutions to 
$e_2=e_4$  (formulae (\ref{seq2}))}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{thebibliography}{99}
\frenchspacing


\bibitem{Bl}
R. E. Block, 
{\it Determination of differentially simple rings with a minimal ideal}, 
Ann. of Math. (2) {\bf 90}~(1969), 433--459.

\bibitem{BlW}
R. E. Block and R. L. Wilson, 
{\it Classification of the restricted simple Lie algebras}, 
J. Algebra {\bf 114}~(1988), 115--259. 

\bibitem{Bo} 
A. Borel, 
{\it Linear Algebraic Groups}, 2nd ed., 
Springer-Verlag, Berlin et al., 1991. 

\bibitem{Br} 
R. Brandl, 
{\it On groups with small Engel depth}, 
Bull. Austral. Math. Soc. {\bf 28} (1983), 101--110.

\bibitem{BN} 
R. Brandl and D. Nikolova, 
{\it Simple groups of small Engel depth}, 
Bull. Austral. Math. Soc. {\bf 33}~(1986), 245--251. 

%\bibitem{BP}
%R. M. Bryant and M. Powell, {\it Two variable laws for 
%$\PSL (2,5)$}, J. Austral. Math. Soc. {\bf 10}~(1969), 499--502.

\bibitem{CMS}
J. Cossey, S. Oates Macdonald, and A. P. Street, 
{\it On the laws of certain finite groups}, 
J. Austral. Math. Soc. {\bf 11}~(1970), 441--489.

\bibitem{DG}
M. Demazure and P. Gabriel, 
{\it Groupes Alg\'ebriques},  
Masson, Paris \& North-Holland, Amsterdam, 1970. 

\bibitem{F} 
P. Flavell, 
{\it Finite groups in which every two elements generate a soluble group}, 
Invent. Math. {\bf 121}~(1995), 279--285.

\bibitem{CG}
C. K. Gupta, 
{\it $2$-metabelian groups}, 
Arch. Math. (Basel) {\bf 19} (1968), 584--587.

\bibitem{G} 
N. D. Gupta, 
{\it Some group laws equivalent to the commutative law}, 
Arch. Math. (Basel) {\bf 17}~(1966), 97--102.

\bibitem{GH} 
N. D. Gupta and H. Heineken, 
{\it Groups with a two-variable commutator identity}, 
Math. Z. {\bf 95}~(1967), 276--287.

\bibitem{HP} 
H. Heineken and P. Neumann, 
{\it Identical relations and decision procedures for groups}, 
J. Austral. Math. Soc. {\bf 7}~(1967), 39--47.

\bibitem{HB}
B. Huppert and N. Blackburn, 
{\it Finite Groups} III, 
Springer-Verlag, Berlin et al., 1982. 
  
\bibitem{J}
G. A. Jones, 
{\it Varieties and simple groups}, 
J. Austral. Math. Soc. {\bf 17}~(1974), 163--173.

\bibitem{L}
S. Lang, 
{\it Algebra}, 3rd ed., Addison-Wesley, 1993.

%\bibitem{Le}
%F. W. Levi, 
%{\it Groups in which the commutator operation satisfies certain algebraic 
%conditions}, 
%J. Indian Math. Soc. (N.S.) {\bf 6} (1942), 87--97.

%\bibitem{M}
%R. P. Martineau, {\it On $2$-modular representations of the 
%Suzuki groups}, Amer. J. Math. {\bf 94}~(1972), 55--72.

\bibitem{BHN} 
B. H. Neumann, 
{\it On a conjecture of Hanna Neumann}, 
Proc. Glasgoq Math. Assoc. {\bf 3} (1956), 13--17. 

\bibitem{Ne}
H. Neumann, 
{\it Varieties of Groups}, 
Springer-Verlag, Berlin et al., 1967.

\bibitem{Nth} 
D. Nikolova, 
{\it Groups with a $2$-variable commutator identity}, 
Ph.~D.~thesis, Sofia Univ., 1983.  

\bibitem{Ni1} 
D. Nikolova, 
{\it Groups with a two-variable commutator identity}, 
C. R. Acad. Bulgare Sci. {\bf 36}~(1983), 721--724.

\bibitem{Ni2} 
D. Nikolova, 
{\it Solubility of finite groups with a two-variable commutator identity}, 
Serdica {\bf 11}~(1985), 59--63. 

%\bibitem{OS}
%S. Oates Macdonald and A. P. Street, {\it On laws in linear 
%groups}, Quart. J. Math. Oxford (2) {\bf 23}~(1972), 1--12.

\bibitem{OP}
S. Oates and M. B. Powell, 
{\it Identical relations in finite groups}, 
J. of Algebra {\bf 1}~(1964), 11--39.


\bibitem{PlR} 
V. P. Platonov and A. S. Rapinchuk, 
{\it Algebraic Groups and Number Theory}, 
Nauka, Moscow, 1991; English transl., Academic Press, Boston et al., 1994.

\bibitem{PPT}
B. Plotkin, E. Plotkin, and A. Tsurkov, 
{\it Geometrical equivalence of groups}, 
Comm. Algebra {\bf 27}~(1999), 4015--4025.

\bibitem{PoR}
A. Potapchik and A. Rapinchuk, 
{\it Normal subgroups of $\SL _{1,D}$ and the classification of finite simple 
groups}, 
Proc. Indian Acad. Sci. (Math. Sci.) {\bf 106}~(1996), 329--368.

\bibitem{Se}
Y. Segev, 
{\it On finite homomorphic images of the multiplicative group of a division 
algebra}, 
Ann. of Math. (2) {\bf 149}~(1999), 219--251.

\bibitem{SS}
Y. Segev and G. Seitz, 
{\it Anisotropic groups of type $A_n$ and the commuting graph of finite 
simple groups}, to appear. 

%\bibitem{So}
%B. Southcott, {\it A basis for the laws of a class of simple 
%groups}, Volumes to the memory of H. Neumann, 1973, pp.500--505.

\bibitem{StF} 
H. Strade and R. Farnsteiner, 
{\it Modular Lie Algebras and Their Representations}, 
Marcel Dekker, New York--Basel, 1988.

\bibitem{T}
J. Thompson, 
{\it Non-solvable finite groups all of whose local subgroups are solvable}, 
Bull. Amer. Math. Soc. {\bf 74}~(1968), 383--437.

\bibitem{W}
B. Weisfeiler, 
{\it On subalgebras of simple Lie algebras of characteristic $p>0$}, 
Trans. Amer. Math. Soc. {\bf 286}~(1984), 471--503. 


\end{thebibliography}


\end{document}


