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\begin{titlepage}
\begin{center}{\Huge \textbf{\textsf{Circle}}}\\[1cm]
\hypersetup{pdftitle=\textsf{Circle}}
\markright{\scriptsize \mbox{}\hfill \textsf{Circle} \hfill\mbox{}}
{\Large \textbf{Adjoint groups of finite radical algebras}}\\[1cm]
{Version 1.1}\\[1cm]
{August 2006}\\[1cm]
\mbox{}\\[2cm]
{\large \textbf{Alexander Konovalov    }}\\
{\large \textbf{Panagiotis Soules   }}\\
\hypersetup{pdfauthor=Alexander Konovalov    ; Panagiotis Soules   }
\end{center}\vfill

\mbox{}\\
{\mbox{}\\
\small \noindent \textbf{Alexander Konovalov    } --- Email: \href{mailto://konovalov@member.ams.org}{\texttt{konovalov@member.ams.org}}\\
 --- Homepage: \href{http://homepages.vub.ac.be/~okonoval/}{\texttt{http://homepages.vub.ac.be/\~{}okonoval/}}\\
 --- Address: \begin{minipage}[t]{8cm}\noindent
 Department of Mathematics\\
 Zaporozhye National University\\
 Zaporozhye, 69063 Ukraine\\
 Current address:\\
 Department of Mathematics\\
 Vrije Universiteit Brussel\\
 Pleinlaan 2, Brussels\\
 B-1050 Belgium \end{minipage}
}\\
{\mbox{}\\
\small \noindent \textbf{Panagiotis Soules   } --- Email: \href{mailto://psoules@math.uoa.gr}{\texttt{psoules@math.uoa.gr}}\\
 --- Address: \begin{minipage}[t]{8cm}\noindent
 Department of Mathematics\\
 National and Capodistrian University of Athens\\
 Panepistimioupolis, GR-15784, Athens, Greece \end{minipage}
}\\
\end{titlepage}

\newpage\setcounter{page}{2}
{\small 
\section*{Abstract}
\logpage{[ 0, 0, 1 ]}
 \index{Circle package@\textsf{Circle} package} The \textsf{GAP}4 package \textsf{Circle} extends the \textsf{GAP} functionality for computations in adjoint groups of associative rings. It
provides functionality to construct circle objects that will respect the
circle multiplication $ r \cdot s = r + s + rs $, create multiplicative groups, generated by such objects, and compute groups
of elements, invertible with respect to this operation, for finite radical
algebras and finite associative rings without one. 

 }\\[1cm]
{\small 
\section*{Copyright}
\logpage{[ 0, 0, 2 ]}
 {\copyright} 2006 by Alexander Konovalov and Panagiotis Soules 

 \textsf{Circle} is free software; you can redistribute it and/or modify it under the terms of
the GNU General Public License as published by the Free Software Foundation;
either version 2 of the License, or (at your option) any later version. For
details, see the FSF's own site \href{http://www.gnu.org/licenses/gpl.html}{\texttt{http://www.gnu.org/licenses/gpl.html}}. 

 If you obtained \textsf{Circle}, we would be grateful for a short notification sent to one of the authors. 

 If you publish a result which was partially obtained with the usage of \textsf{Circle}, please cite it in the following form: 

 A. Konovalov, P. Soules. \emph{Circle --- Adjoint groups of associative rings, Version 1.1;} 2006 (\href{http://homepages.vub.ac.be/~okonoval/circle.htm}{\texttt{http://homepages.vub.ac.be/\~{}okonoval/circle.htm}}). }\\[1cm]
{\small 
\section*{Acknowledgements}
\logpage{[ 0, 0, 3 ]}
 We acknowledge very much Alexander Hulpke for helpful comments and advise. 

 }\\[1cm]
\newpage

\def\contentsname{Contents\logpage{[ 0, 0, 4 ]}}

\tableofcontents
\newpage

  
\chapter{\textcolor{Chapter }{Introduction}}\label{Intro}
\logpage{[ 1, 0, 0 ]}
{
  
\section{\textcolor{Chapter }{General aims}}\label{IntroAbstract}
\logpage{[ 1, 1, 0 ]}
{
  Let $R$ be an associative ring, not necessarily with one. The set of all elements of $R$ forms a monoid with the neutral element $0$ from $R$ under the operation $ r \cdot s = r + s + rs $ defined for all $r$ and $s$ of $R$. This operation is called the \emph{circle multiplication}, and it is also known as the \emph{star multiplication}. The monoid of elements of $R$ under the circle multiplication is called the adjoint semigroup of $R$ and is denoted by $R^{ad}$. The group of all invertible elements of this monoid is called the adjoint
group of $R$ and is denoted by $R^{*}$. 

 These notions naturally lead to a number of questions about the connection
between a ring and its adjoint group, for example, how the ring properties
will determine properties of the adjoint group; which groups can appear as
adjoint groups of rings; which rings can have adjoint groups with prescribed
properties, etc. 

 For example, V. O. Gorlov in \cite{Gorlov-1995} gives a full list of finite nilpotent algebras $R$, such that $R^2 \ne 0$ and the adjoint group of $R$ is metacyclic (but not cyclic). 

 S. V. Popovich and Ya. P. Sysak in \cite{Popovich-Sysak-1997} characterize all quasiregular algebras such that all subgroups of their
adjoint group are their subalgebras. In particular, they show that all
algebras of such type are nilpotent with nilpotency index at most three. 

 Various connections between properties of a ring and its adjoint group were
considered by O. D. Artemovych and Yu. B. Ishchuk in \cite{Artemovych-Ishchuk-1997}. 

 B. Amberg and L. S. Kazarin in \cite{Amberg-Kazarin-2000} give the description of all nonisomorphic finite $p$-groups that can occur as the adjoint group of some nilpotent $p$-algebra of the dimension at most 5. 

 In \cite{Amberg-Sysak-2001} B. Amberg and Ya. P. Sysak give a survey of results on adjoint groups of
radical rings, including such topics as subgroups of the adjoint group;
nilpotent groups which are isomorphic to the adjoint group of some radical
ring; adjoint groups of finite nilpotent $p$-algebras. The authors continued
their investigations in further papers \cite{Amberg-Sysak-2002} and \cite{Amberg-Sysak-2004}. 

 In \cite{Kazarin-Soules-2004} L. S. Kazarin and P. Soules study associative nilpotent algebras over a field
of positive characteristic whose adjoint group has a small number of
generators. 

 The main objective of the proposed \textsf{GAP}4 package \textsf{Circle} is to extend the \textsf{GAP} functionality for computations in adjoint groups of associative rings to make
it possible to use the \textsf{GAP} system for the investigation of the above described questions. 

 \textsf{Circle} provides functionality to construct circle objects that will respect the
circle multiplication $ r \cdot s = r + s + rs $, create multiplicative groups, generated by such objects, and compute groups
of elements, invertible with respect to this operation, for finite radical
algebras and finite associative rings without one. 

 Also we hope that the package will be useful as an example of extending the \textsf{GAP} system with new multiplicative objects. }

  
\section{\textcolor{Chapter }{Installation and system requirements}}\label{IntroInstall}
\logpage{[ 1, 2, 0 ]}
{
  \textsf{Circle} does not use external binaries and, therefore, works without restrictions on
the type of the operating system. It is designed for \textsf{GAP}4.4 and no compatibility with previous releases of \textsf{GAP}4 is guaranteed. 

 To use the \textsf{Circle} online help it is necessary to install the \textsf{GAP}4 package \textsf{GAPDoc} by Frank L\"ubeck and Max Neunh\"offer, which is available from the \textsf{GAP} site or from \href{http://www.math.rwth-aachen.de/~Frank.Luebeck/GAPDoc/}{\texttt{http://www.math.rwth-aachen.de/\~{}Frank.Luebeck/GAPDoc/}}. 

 \textsf{Circle} is distributed in standard formats (\texttt{zoo}, \texttt{tar.gz}, \texttt{tar.bz2}, \texttt{-win.zip}) and can be obtained from \href{http://homepages.vub.ac.be/~okonoval/circle.htm}{\texttt{http://homepages.vub.ac.be/\~{}okonoval/circle.htm}}. To unpack the archive \texttt{circle-1.1.zoo} you need the program \texttt{unzoo}, which can be obtained from the \textsf{GAP} homepage \href{http://www.gap-system.org/}{\texttt{http://www.gap-system.org/}} (see section `Distribution'). To install \textsf{Circle}, copy this archive into the \texttt{pkg} subdirectory of your \textsf{GAP}4.4 installation. The subdirectory \texttt{circle} will be created in the \texttt{pkg} directory after the following command: 

 \texttt{unzoo -x circle-1.1.zoo} 

 }

 }

  
\chapter{\textcolor{Chapter }{\textsf{Circle} functions}}\label{Funct}
\logpage{[ 2, 0, 0 ]}
{
  To use the \textsf{Circle} package first you need to load it as follows: 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> LoadPackage("circle");
  -----------------------------------------------------------------------------
  Loading  Circle 1.0 (Adjoint groups of associative rings)
  by Alexander Konovalov (http://homepages.vub.ac.be/~okonoval/) and
     Panagiotis Soules (psoules@math.uoa.gr).
  -----------------------------------------------------------------------------
  true
  gap>
  
\end{Verbatim}
 
\section{\textcolor{Chapter }{Circle objects}}\label{CircleObjects}
\logpage{[ 2, 1, 0 ]}
{
  Because for elements of the ring $R$ the ordinary multiplication is already denoted by \texttt{*}, for the implementation of the circle multiplication in the adjoint semigroup
we need to wrap up ring elements as CircleObjects, for which \texttt{*} is defined to be the circle multiplication. 

\subsection{\textcolor{Chapter }{CircleObject}}
\logpage{[ 2, 1, 1 ]}\nobreak
{\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{CircleObject( x )\index{CircleObject@\texttt{CircleObject}}
\label{CircleObject}
}\hfill{\scriptsize (attribute)}}\\


 Let \mbox{\texttt{x}} be a ring element. Then \texttt{CircleObject(x)} returns the corresponding circle object. If \mbox{\texttt{x}} lies in the family \texttt{fam}, then \texttt{CircleObject(x)} lies in the family \texttt{CircleFamily} (\ref{CircleFamily}), corresponding to the family \texttt{fam}. 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> x := 2;                                   
  2
  gap> a := CircleObject( x );
  CircleObject( 2 )
  gap> FamilyObj( a ) = CircleFamily( FamilyObj( x ) ); 
  true
  
\end{Verbatim}
 }

 

\subsection{\textcolor{Chapter }{UnderlyingRingElement}}
\logpage{[ 2, 1, 2 ]}\nobreak
{\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{UnderlyingRingElement( x )\index{UnderlyingRingElement@\texttt{UnderlyingRingElement}}
\label{UnderlyingRingElement}
}\hfill{\scriptsize (operation)}}\\


 Let \mbox{\texttt{x}} be a circle object. Then \texttt{UnderlyingRingElement(\mbox{\texttt{x}})} returns the corresponding ring element. 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> x := 2;                                   
  2
  gap> a := CircleObject( x );
  CircleObject( 2 )
  gap> UnderlyingRingElement( a );    
  2
  gap> FamilyObj( a ) = CircleFamily( FamilyObj( x ) ); 
  true
  
\end{Verbatim}
 }

 

\subsection{\textcolor{Chapter }{IsCircleObject}}
\logpage{[ 2, 1, 3 ]}\nobreak
{\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{IsCircleObject( x )\index{IsCircleObject@\texttt{IsCircleObject}}
\label{IsCircleObject}
}\hfill{\scriptsize (Category)}}\\
\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{IsCircleObjectCollection( x )\index{IsCircleObjectCollection@\texttt{IsCircleObjectCollection}}
\label{IsCircleObjectCollection}
}\hfill{\scriptsize (Category)}}\\


 An object \mbox{\texttt{x}} lies in the category \texttt{IsCircleObject} if and only if it lies in a family constructed by \texttt{CircleFamily} (\ref{CircleFamily}). Since circle objects can be multiplied via \texttt{*} with elements in their family, and we want to implement operations \texttt{One} and \texttt{Inverse} to deal with groups they generate, circle objects are implemented in the
category \texttt{IsMultiplicativeElementWithInverse}. 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> IsCircleObject( 2 );             
  false
  gap> IsCircleObject( CircleObject( 2 ) );
  true
  gap> IsMultiplicativeElementWithInverse( CircleObject( 2 ) );
  true
  gap> One( CircleObject( 2 ) );
  CircleObject( 0 )
  gap> CircleObject( -2 )^-1;
  CircleObject( -2 )
  
\end{Verbatim}
 }

 

\subsection{\textcolor{Chapter }{IsPositionalObjectOneSlotRep}}
\logpage{[ 2, 1, 4 ]}\nobreak
{\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{IsPositionalObjectOneSlotRep( x )\index{IsPositionalObjectOneSlotRep@\texttt{IsPositionalObjectOneSlotRep}}
\label{IsPositionalObjectOneSlotRep}
}\hfill{\scriptsize (Representation)}}\\
\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{IsDefaultCircleObject( x )\index{IsDefaultCircleObject@\texttt{IsDefaultCircleObject}}
\label{IsDefaultCircleObject}
}\hfill{\scriptsize (Representation)}}\\


 To store the corresponding circle object, we need only to store the underlying
ring element. Since this is quite common situation, we defined the
representation \texttt{IsPositionalObjectOneSlotRep} for a more general case. Then we defined \texttt{IsDefaultCircleObject} as a synonym of \texttt{IsPositionalObjectOneSlotRep} for objects in \texttt{IsCircleObject} (\ref{IsCircleObject}). 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> IsPositionalObjectOneSlotRep( CircleObject( 2 ) );
  true
  gap> IsDefaultCircleObject( CircleObject( 2 ) );                          
  true
  
\end{Verbatim}
 }

 

\subsection{\textcolor{Chapter }{CircleFamily}}
\logpage{[ 2, 1, 5 ]}\nobreak
{\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{CircleFamily( fam )\index{CircleFamily@\texttt{CircleFamily}}
\label{CircleFamily}
}\hfill{\scriptsize (attribute)}}\\


 \texttt{CircleFamily(fam)} is a family, elements of which are in one-to-one correspondence with elements
of the family \mbox{\texttt{fam}}, but with the circle multiplication as an infix multiplication. That is, for $x$, $y$ in \mbox{\texttt{fam}}, the product of their images in the \texttt{CircleFamily(fam)} will be the image of $ x + y + x y $. 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> x:=CircleObject(2)*CircleObject(3);                       
  CircleObject( 11 )
  gap> y:=CircleObject(2+3+2*3);
  CircleObject( 11 )
  gap> x=y;
  true
  gap> FamilyObj(x)=FamilyObj(y);                                                                 
  true
  
\end{Verbatim}
 }

 }

 
\section{\textcolor{Chapter }{Operations with circle objects}}\label{CircleOperations}
\logpage{[ 2, 2, 0 ]}
{
  

\subsection{\textcolor{Chapter }{OneOp}}
\logpage{[ 2, 2, 1 ]}\nobreak
{\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{OneOp( x )\index{OneOp@\texttt{OneOp}}
\label{OneOp}
}\hfill{\scriptsize (operation)}}\\


 This operation returns the multiplicative neutral element for the circle
object \mbox{\texttt{x}}. The result is the circle object corresponding to the additive neutral
element of the appropriate ring. 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> One( CircleObject( 5 ) );
  CircleObject( 0 )
  gap> One( CircleObject( 5 ) ) = CircleObject( Zero( 5 ) );
  true
  gap> One( CircleObject( [ [ 1, 1 ],[ 0, 1 ] ] ) );
  CircleObject( [ [ 0, 0 ], [ 0, 0 ] ] )
  
\end{Verbatim}
 }

 \pagebreak 

\subsection{\textcolor{Chapter }{InverseOp}}
\logpage{[ 2, 2, 2 ]}\nobreak
{\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{InverseOp( x )\index{InverseOp@\texttt{InverseOp}}
\label{InverseOp}
}\hfill{\scriptsize (operation)}}\\


 For a circle object \mbox{\texttt{x}}, returns the multiplicative inverse of \mbox{\texttt{x}} with respect to the circle multiplication; if such one does not exist then \texttt{fail} is returned. 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> CircleObject( -2 )^-1;                        
  CircleObject( -2 )
  gap> CircleObject( 2 )^-1; 
  CircleObject( -2/3 )
  gap> CircleObject( -2 )*CircleObject( -2 )^-1;
  CircleObject( 0 )
  gap> m := CircleObject( [ [ 1, 1 ], [ 0, 1 ] ] );   
  CircleObject( [ [ 1, 1 ], [ 0, 1 ] ] )
  gap> m^-1;    
  CircleObject( [ [ -1/2, -1/4 ], [ 0, -1/2 ] ] )
  gap> m * m^-1;
  CircleObject( [ [ 0, 0 ], [ 0, 0 ] ] )
  gap> CircleObject( [ [ 0, 1 ], [ 1, 0 ] ] )^-1; 
  fail
  
\end{Verbatim}
 }

 

\subsection{\textcolor{Chapter }{IsUnit}}
\logpage{[ 2, 2, 3 ]}\nobreak
{\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{IsUnit( R, x )\index{IsUnit@\texttt{IsUnit}}
\label{IsUnit}
}\hfill{\scriptsize (operation)}}\\
\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{IsUnit( x )\index{IsUnit@\texttt{IsUnit}}
\label{IsUnit}
}\hfill{\scriptsize (operation)}}\\


 Let \mbox{\texttt{x}} be a circle object corresponding to an element of the ring \mbox{\texttt{R}}. Then the operation \texttt{IsUnit} returns \texttt{true}, if \mbox{\texttt{x}} is invertible in \mbox{\texttt{R}} with respect to the circle multiplication, and \texttt{false} otherwise. 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> IsUnit( Integers, CircleObject( -2 ) );
  true
  gap> IsUnit( Integers, CircleObject( 2 ) ); 
  false
  gap> IsUnit( Rationals, CircleObject( 2 ) );        
  true
  gap> IsUnit( ZmodnZ(8), CircleObject( ZmodnZObj(2,8) ) );
  true
  gap> m := CircleObject( [ [ 1, 1 ],[ 0, 1 ] ] );
  CircleObject( [ [ 1, 1 ], [ 0, 1 ] ] )
  gap> IsUnit( FullMatrixAlgebra( Rationals, 2 ), m );
  true
  
\end{Verbatim}
 \pagebreak In the second form the result will be returned with respect to the default
ring of the circle object \mbox{\texttt{x}}. 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> IsUnit( CircleObject( -2 ) );
  true
  gap> IsUnit( CircleObject( 2 ) ); 
  false
  gap> IsUnit( CircleObject( ZmodnZObj(2,8) ) );
  true
  gap> IsUnit( CircleObject( [ [ 1, 1 ],[ 0, 1 ] ] ) );                                    
  true
  
\end{Verbatim}
 }

 

\subsection{\textcolor{Chapter }{IsCircleUnit}}
\logpage{[ 2, 2, 4 ]}\nobreak
{\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{IsCircleUnit( R, x )\index{IsCircleUnit@\texttt{IsCircleUnit}}
\label{IsCircleUnit}
}\hfill{\scriptsize (operation)}}\\
\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{IsCircleUnit( x )\index{IsCircleUnit@\texttt{IsCircleUnit}}
\label{IsCircleUnit}
}\hfill{\scriptsize (operation)}}\\


 Let \mbox{\texttt{x}} be an element of the ring \mbox{\texttt{R}}. Then \texttt{IsCircleUnit( R, x )} determines whether \mbox{\texttt{x}} is invertible in \mbox{\texttt{R}} with respect to the circle multilpication. This is equivalent to the condition
that 1+\mbox{\texttt{x}} is a unit in \mbox{\texttt{R}} with respect to the ordinary multiplication. 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> IsCircleUnit( Integers, -2 );
  true
  gap> IsCircleUnit( Integers, 2 ); 
  false
  gap> IsCircleUnit( Rationals, 2 );          
  true
  gap> IsCircleUnit( ZmodnZ(8), ZmodnZObj(2,8) ); 
  true
  gap> m := [ [ 1, 1 ],[ 0, 1 ] ];                
  [ [ 1, 1 ], [ 0, 1 ] ]
  gap> IsCircleUnit( FullMatrixAlgebra(Rationals,2), m );
  true
  
\end{Verbatim}
 In the second form the result will be returned with respect to the default
ring of \mbox{\texttt{x}}. 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> IsCircleUnit( -2 );                               
  true
  gap> IsCircleUnit( 2 ); 
  false
  gap> IsCircleUnit( ZmodnZObj(2,8) );           
  true
  gap> IsCircleUnit( [ [ 1, 1 ],[ 0, 1 ] ] ); 
  true
  
\end{Verbatim}
 }

 }

 \pagebreak 
\section{\textcolor{Chapter }{Construction of the adjoint group}}\label{CircleAdjointGroups}
\logpage{[ 2, 3, 0 ]}
{
  

\subsection{\textcolor{Chapter }{AdjointGroup}}
\logpage{[ 2, 3, 1 ]}\nobreak
{\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{AdjointGroup( R )\index{AdjointGroup@\texttt{AdjointGroup}}
\label{AdjointGroup}
}\hfill{\scriptsize (attribute)}}\\


 If \mbox{\texttt{R}} is a finite radical algebra then \texttt{AdjointGroup(\mbox{\texttt{R}})} will return the adjoint group of \mbox{\texttt{R}}, given as a group generated by a set of circle objects. 

 To compute the adjoint group of a finite radical algebra, \textsf{Circle} uses the fact that all elements of a radical algebra form a group with respect
to the circle multiplication. Thus, the adjoint group of \mbox{\texttt{R}} coincides with \mbox{\texttt{R}} elementwise, and we can randomly select an appropriate set of generators for
the adjoint group. 

 The warning is displayed by \texttt{IsGeneratorsOfMagmaWithInverses} method defined in \texttt{gap4r4/lib/grp.gi} and may be ignored. 

 \textsc{WARNINGS:} 

 1. The set of generators of the returned group is not required to be the
minimal generating set. 

 2. \texttt{AdjointGroup} is stored as an attribute of \mbox{\texttt{R}}, so for the same copy of \mbox{\texttt{R}} calling it again you will get the same result. But if you will create another
copy of \mbox{\texttt{R}} in the future, the output may differ because of the random selection of
generators. If you want to have the same generating set, next time you should
construct a group immediately specifying circle objects that generate it. 

 3. In most cases, to investigate some properties of the adjoint group, it is
necessary first to convert it to an isomorphic permutation group or to a
PcGroup. 

 For example, we can create the following commutative 2-dimensional radical
algebra of order 4 over the field of two elements, and show that its adjoint
group is a cyclic group of order 4: 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> x:=[ [ 0, 1, 0 ],
  >         [ 0, 0, 1 ],
  >         [ 0, 0, 0 ] ];;
  gap> R := Algebra( GF(2), [ One(GF(2))*x ] );  
  <algebra over GF(2), with 1 generators>
  gap> RadicalOfAlgebra( R ) = R;
  true
  gap> Dimension(R);
  2
  gap> G := AdjointGroup( R );    
  #I  default `IsGeneratorsOfMagmaWithInverses' method returns `true' for 
  [ CircleObject( [ [ 0*Z(2), 0*Z(2), Z(2)^0 ], [ 0*Z(2), 0*Z(2), 0*Z(2) ], 
        [ 0*Z(2), 0*Z(2), 0*Z(2) ] ] ) ]
  <group of size 4 with 2 generators>
  gap> Size( R ) = Size( G );
  true
  gap> StructureDescription( G );
  "C4"
  
\end{Verbatim}
 \pagebreak In the following example we construct a non-commutative 3-dimensional radical
algebra of order 8 over the field of two elements, and demonstrate that its
adjoint group is the dihedral group of order 8: 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> x:=[ [ 0, 1, 0 ],
  >         [ 0, 0, 0 ],     
  >         [ 0, 0, 0 ] ];;
  gap> y:=[ [ 0, 0, 0 ],     
  >         [ 0, 0, 1 ],  
  >         [ 0, 0, 0 ] ];;
  gap> R := Algebra( GF(2), One(GF(2))*[x,y] );  
  <algebra over GF(2), with 2 generators>
  gap> RadicalOfAlgebra(R) = R;                
  true
  gap> Dimension(R);
  3
  gap> G := AdjointGroup( R );
  #I  default `IsGeneratorsOfMagmaWithInverses' method returns `true' for 
  [ CircleObject( [ [ 0*Z(2), Z(2)^0, Z(2)^0 ], [ 0*Z(2), 0*Z(2), Z(2)^0 ], 
        [ 0*Z(2), 0*Z(2), 0*Z(2) ] ] ) ]
  <group of size 8 with 2 generators>
  gap> StructureDescription( G );
  "D8"
  
\end{Verbatim}
 If the ring \mbox{\texttt{R}} is not a radical algebra, but also is not a ring with one, then \textsf{Circle} will use another approach. We will enumerate all elements of the ring \mbox{\texttt{R}} and select those that are units with respect to the circle multiplication.
Then we will use a random approach similar to the case of the radical algebra,
to find some generating set of the adjoint group. Again, all warnings 1-3
above refer also to this case. 

 Of course, enumeration of all elements of \mbox{\texttt{R}} should be feasible for this computation. In the following example we
demonstrate how it works for rings, generated by residue classes: 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> R := Ring( [ ZmodnZObj(2,8) ] );
  <ring with 1 generators>
  gap> G := AdjointGroup( R );
  #I  default `IsGeneratorsOfMagmaWithInverses' method returns `true' for 
  [ CircleObject( ZmodnZObj( 2, 8 ) ) ]
  <group of size 4 with 2 generators>
  gap> StructureDescription( G );
  "C2 x C2"
  gap> R := Ring( [ ZmodnZObj(2,256) ] );   
  <ring with 1 generators>
  gap> G := AdjointGroup( R );
  #I  default `IsGeneratorsOfMagmaWithInverses' method returns `true' for 
  [ CircleObject( ZmodnZObj( 234, 256 ) ) ]
  <group of size 128 with 2 generators>
  gap> StructureDescription( G );
  "C64 x C2"
  
\end{Verbatim}
 \pagebreak If \mbox{\texttt{R}} has a unity $1$, then the set $1+R^{ad}$, where $R^{ad}$ is the adjoint semigroup of \mbox{\texttt{R}}, coincides with the multiplicative semigroup $R^{mult}$ of $R$, and the map $ r \mapsto (1+r) $ for $r$ in $R$ is an isomorphism from $R^{ad}$ onto $R^{mult}$. 

 Similarly, the set $1+R^*$, where $R^{*}$ is the adjoint group of \mbox{\texttt{R}}, coincides with the unit group of $R$, which we denote $U(R)$, and the map $r \mapsto (1+r)$ for $r$ in $R$ is an isomorphism from $R^*$ onto $U(R)$. 

 We demonstrate this isomorphism using the following example. 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> FG := GroupRing( GF(2), DihedralGroup(8) );
  <algebra-with-one over GF(2), with 3 generators>
  gap> R := AugmentationIdeal( FG );
  <two-sided ideal in <algebra-with-one over GF(2), with 3 generators>, 
    (dimension 7)>
  gap> G := AdjointGroup( R );
  #I  default `IsGeneratorsOfMagmaWithInverses' method returns `true' for 
  [ CircleObject( (Z(2)^0)*f2+(Z(2)^0)*f1*f2 ) ]
  <group of size 128 with 4 generators>
  gap> IdGroup( G );
  [ 128, 170 ]
  gap> IdGroup( Units( FG ) );
  #I  LAGUNA package: Computing the unit group ...
  [ 128, 170 ]
  
\end{Verbatim}
 This is is why we did not implemented in \textsf{Circle} adjoint groups for associative rings with one. If you will try to compute the
adjoint group for such a ring, you will get an error message telling that you
can investigate the unit group of the ring instead. 

 If \mbox{\texttt{R}} is infinite, an error message will appear, telling that \textsf{Circle} does not provide methods to deal with infinite rings. }

 }

 \pagebreak  
\section{\textcolor{Chapter }{Service functions}}\label{Service}
\logpage{[ 2, 4, 0 ]}
{
  

\subsection{\textcolor{Chapter }{InfoCircle}}
\logpage{[ 2, 4, 1 ]}\nobreak
{\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{InfoCircle\index{InfoCircle@\texttt{InfoCircle}}
\label{InfoCircle}
}\hfill{\scriptsize (info class)}}\\


 \texttt{InfoCircle} is a special Info class for \textsf{Circle} algorithms. It has 2 levels: 0 (default) and 1. To change info level to \texttt{k}, use command \texttt{SetInfoLevel(InfoCircle, k)}. 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> SetInfoLevel( InfoCircle, 1 );
  gap> SetInfoLevel(InfoCircle,1);
  gap> R := Ring( [ ZmodnZObj(2,8) ]);
  <ring with 1 generators>
  gap> G := AdjointGroup( R );
  #I  Circle : <R> is not a radical algebra, computing circle units ...
  #I  Circle : searching generators for adjoint group ...
  #I  default `IsGeneratorsOfMagmaWithInverses' method returns `true' for 
  [ CircleObject( ZmodnZObj( 6, 8 ) ) ]
  <group of size 4 with 2 generators>
  gap> SetInfoLevel( InfoCircle, 0 );
  
\end{Verbatim}
 }

 

\subsection{\textcolor{Chapter }{CIRCLEBuildManual}}
\logpage{[ 2, 4, 2 ]}\nobreak
{\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{CIRCLEBuildManual(  )\index{CIRCLEBuildManual@\texttt{CIRCLEBuildManual}}
\label{CIRCLEBuildManual}
}\hfill{\scriptsize (function)}}\\


 This function is used to build the manual in the following formats: DVI, PDF,
PS, HTML and text for online help. We recommend that the user should have a
recent and fairly complete {\TeX} distribution. Since \textsf{Circle} is distributed together with its manual, it is not necessary for the user to
use this function. Normally it is intended to be used by the developers only.
This is the only function of \textsf{Circle} which requires UNIX/Linux environment. }

 

\subsection{\textcolor{Chapter }{CIRCLEBuildManualHTML}}
\logpage{[ 2, 4, 3 ]}\nobreak
{\noindent\textcolor{FuncColor}{$\Diamond$\ \texttt{CIRCLEBuildManualHTML(  )\index{CIRCLEBuildManualHTML@\texttt{CIRCLEBuildManualHTML}}
\label{CIRCLEBuildManualHTML}
}\hfill{\scriptsize (function)}}\\


 This fuction is used to build the manual only in HTML format. This does not
depend on the availability of the {\TeX} installation and works under Windows and MacOS as well. Since \textsf{Circle} is distributed together with its manual, it is not necessary for the user to
use this function. Normally it is intended to be used by the developers only. }

 }

 }

  
\chapter{\textcolor{Chapter }{A sample computation with \textsf{Circle}}}\label{Example}
\logpage{[ 3, 0, 0 ]}
{
  Here we give an example to give the reader an idea what \textsf{Circle} is able to compute. 

 It was proved in \cite{Kazarin-Soules-2004} that if $R$ is a finite nilpotent two-generated algebra over a field of characteristic $p>3$ whose adjoint group has at most three generators, then the dimension of $R$ is not greater than 9. Also, an example of the 6-dimensional such algebra with
the 3-generated adjoint group was given there. We will construct the algebra
from this example and investigate it using \textsf{Circle}. First we create two matrices that determine its generators: 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
                                                                                            
  gap> x:=[ [ 0, 1, 0, 0, 0, 0, 0 ],
  >         [ 0, 0, 0, 1, 0, 0, 0 ],
  >         [ 0, 0, 0, 0, 1, 0, 0 ],
  >         [ 0, 0, 0, 0, 0, 0, 1 ],
  >         [ 0, 0, 0, 0, 0, 1, 0 ],
  >         [ 0, 0, 0, 0, 0, 0, 0 ],
  >         [ 0, 0, 0, 0, 0, 0, 0 ] ];;
  gap> y:=[ [ 0, 0, 1, 0, 0, 0, 0 ],
  >         [ 0, 0, 0, 0,-1, 0, 0 ],
  >         [ 0, 0, 0, 1, 0, 1, 0 ],
  >         [ 0, 0, 0, 0, 0, 1, 0 ],
  >         [ 0, 0, 0, 0, 0, 0,-1 ],
  >         [ 0, 0, 0, 0, 0, 0, 0 ],
  >         [ 0, 0, 0, 0, 0, 0, 0 ] ];;
                                                                                                  
\end{Verbatim}
 Now we construct this algebra in characteristic five and check its basic
properties: 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
                                                                                            
  gap> R := Algebra( GF(5), One(GF(5))*[x,y] );
  <algebra over GF(5), with 2 generators>
  gap> Dimension( R );
  6
  gap> Size( R );
  15625
  gap> RadicalOfAlgebra( R ) = R;
  true
                                                                                                  
\end{Verbatim}
 Then we compute the adjoint group of \texttt{R}. During the computation a warning will be displayed. It is caused by the
method for \texttt{IsGeneratorsOfMagmaWithInverses} defined in the file \texttt{gap4r4/lib/grp.gi} from the \textsf{GAP} library, and may be safely ignored. 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
   
  gap> G := AdjointGroup( R );
  #I  default `IsGeneratorsOfMagmaWithInverses' method returns `true' for 
  [ CircleObject( [ [ 0*Z(5), Z(5), Z(5), Z(5)^3, Z(5), 0*Z(5), Z(5)^2 ],
        [ 0*Z(5), 0*Z(5), 0*Z(5), Z(5), Z(5)^3, Z(5)^3, Z(5)^3 ],
        [ 0*Z(5), 0*Z(5), 0*Z(5), Z(5), Z(5), 0*Z(5), Z(5) ],
        [ 0*Z(5), 0*Z(5), 0*Z(5), 0*Z(5), 0*Z(5), Z(5), Z(5) ],
        [ 0*Z(5), 0*Z(5), 0*Z(5), 0*Z(5), 0*Z(5), Z(5), Z(5)^3 ],
        [ 0*Z(5), 0*Z(5), 0*Z(5), 0*Z(5), 0*Z(5), 0*Z(5), 0*Z(5) ],
        [ 0*Z(5), 0*Z(5), 0*Z(5), 0*Z(5), 0*Z(5), 0*Z(5), 0*Z(5) ] ] ) ]
  <group of size 15625 with 3 generators>
                                                                                                  
\end{Verbatim}
 Now we can find the minimal generating set of \texttt{G} and check that \texttt{G} it is 3-generated. To do this, first we need to convert it to the isomorphic
PcGroup: 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
   
  gap> f := IsomorphismPcGroup( G );;
  gap> H := Image( f );
  Group([ f1, f2, f3, f4, f5, f6 ])
  gap> gens := MinimalGeneratingSet( H );
  [ f1, f2, f5 ]
  gap> gens:=List( gens, x -> UnderlyingRingElement(PreImage(f,x)));;
  gap> Perform(gens,Display);                                        
   . 3 3 4 4 . 1
   . . . 3 2 1 4
   . . . 3 3 2 4
   . . . . . 3 3
   . . . . . 3 2
   . . . . . . .
   . . . . . . .
   . 3 1 1 . . .
   . . . 3 4 . 1
   . . . 1 3 2 .
   . . . . . 1 3
   . . . . . 3 4
   . . . . . . .
   . . . . . . .
   . 2 2 3 2 . 4
   . . . 2 3 3 3
   . . . 2 2 . 2
   . . . . . 2 2
   . . . . . 2 3
   . . . . . . .
   . . . . . . .
                                                                                                  
\end{Verbatim}
 It appears that the adjoint group of the algebra from example will be
3-generated in characteristic three as well: 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> R := Algebra( GF(3), One(GF(3))*[x,y] );
  <algebra over GF(3), with 2 generators>
  gap> G := AdjointGroup( R );
  #I  default `IsGeneratorsOfMagmaWithInverses' method returns `true' for 
  [ CircleObject( [ [ 0*Z(3), 0*Z(3), Z(3)^0, Z(3)^0, Z(3), Z(3), 0*Z(3) ],
        [ 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), Z(3), Z(3)^0, Z(3)^0 ],
        [ 0*Z(3), 0*Z(3), 0*Z(3), Z(3)^0, 0*Z(3), Z(3), Z(3) ],
        [ 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), Z(3)^0, 0*Z(3) ],
        [ 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), Z(3) ],
        [ 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3) ],
        [ 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3) ] ] ) ]
  <group of size 729 with 3 generators>
  gap> H := Image( IsomorphismPcGroup( G ) );
  Group([ f1, f2, f3, f4, f5, f6 ])
  gap> MinimalGeneratingSet( H );
  [ f1, f2, f4 ]
                                                                                                  
\end{Verbatim}
 But this is not the case in characteristic two, where the adjoint group is
4-generated: 
\begin{Verbatim}[fontsize=\small,frame=single,label=Example]
  
  gap> R := Algebra( GF(2), One(GF(2))*[x,y] );
  <algebra over GF(2), with 2 generators>
  gap> G := AdjointGroup( R );                   
  #I  default `IsGeneratorsOfMagmaWithInverses' method returns `true' for 
  [ CircleObject( [ [ 0*Z(2), Z(2)^0, Z(2)^0, 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2) ],
        [ 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0, Z(2)^0, Z(2)^0, 0*Z(2) ],
        [ 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0, Z(2)^0, Z(2)^0, Z(2)^0 ],
        [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0, Z(2)^0 ],
        [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0, Z(2)^0 ],
        [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ],
        [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ] ] ) ]
  <group of size 64 with 4 generators>
  gap> H := Image( IsomorphismPcGroup( G ) );
  Group([ f1, f2, f3, f4, f5, f6 ])
  gap> MinimalGeneratingSet( H );
  [ f1, f2, f4, f5 ]
                                                                                                  
\end{Verbatim}
 }

 \def\bibname{References\logpage{[ "Bib", 0, 0 ]}}

\bibliographystyle{alpha}
\bibliography{manual}

\def\indexname{Index\logpage{[ "Ind", 0, 0 ]}}


\printindex

\newpage
\immediate\write\pagenrlog{["End"], \arabic{page}];}
\immediate\closeout\pagenrlog
\end{document}
